Why should a clock be "accurate"? Having read that atomic clocks are more accurate than mechanical clocks as they lose a second only in millions of years, I wonder why it is necessary for a reference clock to worry about this, if the definition of the second itself is a function of the number of ticks the clock makes.
Why don't we just use a single simple mechanical clock somewhere with a wound up spring that makes it tick, and whenever it makes a tick, treat it as a second having elapsed?
(Assuming this clock was broadcasting its time via internet ntp servers to everyone in the world)
 A: 
why it is necessary for a reference clock to worry about this, if the definition of the second itself is a function of the number of ticks the clock makes.

The concern is that somebody else (say a scientist in France or China or Botswana) needs to be able to build a clock that measures seconds at the same rate mine does.
If we both have atomic clocks, we can keep our clocks syncronized to within microseconds per year. If we have mechanical clocks they might be different from each other by a second (or anyway some milliseconds) by the end of a year. If we're doing very exact measurements (comparing the arrival times of gamma rays from astronomical events at different parts of the Earth, or just using a GPS navigation system) then a few milliseconds (or even microseconds)  can make a difference in our results.
A: Time doesn't flow, nor is it perceived, according to the ticking of a clock. If you boil an egg while watching a clock that runs slow, you're going to overcook it, regardless of the fact that the clock says you cooked it for exactly the intended duration. "Boil an egg for 10 minutes" is not a useful instruction if the actual duration of 10 minutes is not constant.
A wind-up mechanical clock is not terribly precise and can produce "seconds" of different durations depending on environmental factors like temperature, humidity, etc. If your clock doesn't have a constant tick rate, an egg cooked for "10 minutes" may be overcooked or undercooked, since that same "10 minutes" can represent a variable amount of time.
We need to know that 10 minutes measured today is the same as 10 minutes tomorrow. A mechanical reference clock could slow down over time, resulting in constant-duration processes appearing to take less time - in 100 years, you might find that a perfectly cooked egg only takes 5 "minutes" according to your slowed clock, when in reality, it's the exact same duration.
A: I am adding this answer because I feel like the other answer do not cover some main parts as to really why we a destined to create more and more accurate clocks: more accurate distance calculations and redefining the second itself.
The answer to your question is relativity, in that in your example, a clock can only be accurate relatively (to another one).

Atomic clocks are so accurate that they will lose one second approximately every 100 million years; for reference, the average quartz clock will lose one second every couple of years. On the other hand, Ye’s optical lattice clock will lose one second every 15 billion years, making it the world’s most accurate clock.

https://www.labroots.com/trending/chemistry-and-physics/21278/world-s-accurate-clock-2
As there is no absolute time, there is no absolutely accurate clock either. We happen to live in a universe that is fundamentally quantum mechanical, and probabilistic. All clocks will "lose" some time (a second for example) eventually, the question is only when. But to know that the clock has "lost" a second, you need to compare it to another one.
The point of this whole thing is that the clock can repeat ticks (quantum mechanical processes) with equal time intervals, that is, the ticks are equal. But, since we are talking about quantum mechanical processes, one of the ticks will be of different length from another tick (and these differences will eventually add up to a measurable interval, like a second).

The timekeeping accuracy of an atomic clock is important because the smaller the error in a time measurement, the smaller the error in distance obtained by multiplying the time by the speed of light is.
As engineers make more precise clocks, they are beginning to develop different ytterbium and strontium based types that measure time to {\displaystyle 10^{-18}}{\displaystyle 10^{-18}} seconds. Sometime during the next 10 years these optical clocks are expected to lead to a redefinition of the second, possibly with the Rydberg constant.

https://en.wikipedia.org/wiki/Atomic_clock
So as you can see there are two main reasons we are trying to reach more accuracy:

*

*the smaller the error in time measurement, the smaller the error in distance calculations (because of the absoluteness of the speed of light in the calculations using for example EM waves)


*redefining the second itself
But there is no absolute accuracy, we will always be able to create more and more accurate clocks, and the answer to your question is that this accuracy is always relative.
A: For most of human history, we had a single mechanical clock: the spinning Earth.
Well, actually two mechanical clocks.  The Earth’s spin rate is a good constant, but it’s tricky to measure directly.  The thing that’s easy to measure is the interval between sunrises, but that gets longer and shorter from time to time.  When sunrises are getting farther apart, the weather tends to be getting warmer, so noticing this was useful for agriculture.  That’s because of the second mechanical clock: Earth’s orbit around the Sun.
Synchronizing the daily clock and the annual clock was a terrifically hard problem: the mismatch between a tropical year and a 365-day year is small over the duration of a human life. The solution was the Gregorian calendar, which was invented in the 1500s but not adopted worldwide until the twentieth century.
You don’t care about accuracy if you only have one clock.  It’s impossible to care: you can’t test the accuracy of a single clock.  But if you have ten clocks, you can ask whether they all keep the same time, or whether they all diverge from each other, or whether eight of them stay together but two of them run slow.
A: Most of the answers talk about being able to compare clocks, which is important, but not the fundamental issue. The point of having an accurate clock is to have an accurate and universal measurement for the flow of time.
If you measure a time of 9.58 seconds for Usain Bolt to run 100 meters in 2009 and then measure 9.63 seconds in 2012, does that mean that he got slower or does it mean that the mechanical properties of your junky wind-up clock changed?
If you measure the period of a pulsar to be 1.337 s in 1967 and measure a different period in 2022, did the pulsar change or did your clock change?
Let's say you insist on using your junky wind-up clock, which is located in say, Greenwich, England, and have found a way to broadcast its ticks all over the globe at infinite speeds (I'll even ignore the complication of the finite speed of light). Let's even say your wind-up clock is pretty consistent, but its period varies slightly with temperature. You may find that Mr. Bolt's performance in Berlin or Beijing depends on the current weather in Greenwich. If you measure the period of a pulsar in February of 2022 and then measure it again July of 2022, you will find a slight shift in the period due to the temperature differences in Greenwich between these times. So when comparing measurements of pulsars, moon orbits, LC circuits, and Olympic sprinters, you will have to consult the table of daily temperatures in Greenwich, England.
All of your equations for celestial mechanics and electronics will require terms accounting for the local weather in Greenwich at the time when the measurement was made. This would make any calculations that somehow involve time much more complicated.
You can say, WaterMolecule, the theory of relativity says that all time measurements are relative to the observer, so I can use any old clock. But that is not what relativity says. While an alien spacecraft flying by the Earth at half the speed of light would measure a longer time for Usain Bolt's record breaking sprint (11.06 s), assuming that they have an accurate clock, they would be able to calculate the proper time in stadium rest frame to be 9.58 seconds. Relativity does not obviate the need for an accurate clock.
Time has emerged as a fundamental physical parameter in theory and in experiments because pendulums with the same lengths and LC circuits with the same components yield similar periods wherever the experiments are performed and don't depend on the conditions in some faraway place. If time were subject to random effects due to local conditions in Greenwich, England or at the surface of Betelgeuse, we wouldn't use it as a fundamental parameter. We invented the concept of time specifically because it allows for repeatable experiments. If the local temperature at the surface of Betelgeuse defined the flow of time, physics would have never been invented and we would have little ability to understand the world around us.
A: Let's say we live in Plato's ideal city-state and I am the Philosopher King. I proclaim my sleeping cycle as the clock. One unit of time is the time interval between two consecutive instances of me waking up. Equipped with this clock, you go on the quest to observe nature and understand its patterns. The world would look incredibly confusing and you would not be able to notice any discernable patterns in its behavior. Sometimes two sunrises would happen in one unit of time-interval and sometimes none, sometimes you'd feel hungry six times in one unit of time-interval and sometimes only once, sometimes you'd be able to finish a given amount of work in one unit of time-interval whereas sometimes it would take you multiple units of time-interval to finish the same amount of work even if you're working in the same manner, etc.
You'd soon realize that if you instead use the Sun as your clock and use its two consecutive rises to define the unit interval of time, a lot of things would start looking more robust, more predictable. You'd see that you almost always feel hungry three times during one unit of time-interval, you always finish roughly the same amount of work in every unit of time-interval, etc.

The point is that a clock needs to be a mechanism that is reliably periodic, ideally, perfectly periodic. As you can see, this is circular but circularity is the wrong thing to focus on. The validity of the scheme comes from the fact that as I illustrated in my example above, there are wrong answers to what you proclaim as periodic in that they won't be useful in finding discernable patterns in the universe. Furthermore, to a certain extent, you can argue that you have good reason to proclaim a system periodic even before specifying how to measure time via appealing to symmetry. For example, you can say that the amount of time that passes during a simple pendulum's left-to-right swing must be equal to the amount of time that passes during the same pendulum's right-to-left swing. Of course, this proclamation of periodicity won't be good enough over a large number of oscillations of the pendulum. You'd notice this in the same way that you noticed that my sleeping pattern wasn't reliably periodic. And you'd go on to find an even more reliably periodic system.
So, the quest of finding more and more accurate clocks is the quest of going closer and closer to an ideal periodic system and this is important because the usefulness of the concept of time is not a penny more than the robustness of the periodicity of the clock that is used to define a unit of time.
A: One application for the need of accurate timekeeping would be to evaluate if the fundamental constants really are constant. See for example this article, chapter 6:
https://www.ptb.de/cms/fileadmin/internet/fachabteilungen/abteilung_4/4.4_zeit_und_frequenz/pdf/2010-Peik_NuclPhysB_Fundamentals_constants.pdf
Exact timekeeping also allows for the exact measurement of SI units that are derived from the Unit second. Examples are the metre, the kilogram and the kelvin. Their precision, so to say, relies (amongst other values) on the precision of the second.
A: You could use a standard pendulum, whose period varying over time is known. It would be difficult though to reproduce this exact standard pendulum at every location and do experiments with it. Like every reproduction of a standard meter would vary to some extent
they would always vary to some extent. Most clocks vary in their period times. That's why the caesium clock is made the standard. It varies the least (only a second every million years, where that million years is measured by an ideal non-existent, perfect clock in the mind; one with constant period).
It is in fact the ticking of this caesium clock that is used to broadcast the "right" time over the media. You could use grandma's pendulum clock, but in modern society this could give difficulties.
For example, if a technical process required accurate timing, grandma's pendulum might be causing system failures, explosions, airplanes crashing, and maybe even a third world war. If grandma would have known that she wouldn't have donated it...
A: 
if the definition of the second itself is a function of the number of ticks the clock makes.


Why don't we just use a single simple mechanical clock somewhere with a wound up spring that makes it tick, and whenever it makes a tick, treat it as a second having elapsed?

There is a misconception here. Seconds do not define time nor do clocks define the passage of time. The universe defines the passage of time. Remember that the clock is being used to measure time. The clock doesn't define time. In other words, we are trying to track "universe's clock" which does define the passage of time. Our definition of the second is just to quantify time.
Implicit in this accuracy/precision is that each cycle of the clock is as identical as possible from moment to moment (or rather, tracks the universe's passage of time which is where time dilation gets gnarly). That's what really matters. Not so much the losing one second every million years. That's secondary.
Really, the precision of the clock comes before the accuracy. That is most important: How repeatable each interval is. The accuracy only comes into play when you have a standard you're trying to aspire to such as the theoretical definition of a second or another time standard, or other clocks.

Why don't we just use a single simple mechanical clock somewhere with a wound up spring that makes it tick, and whenever it makes a tick, treat it as a second having elapsed?


(Assuming this clock was broadcasting its time via internet ntp servers to everyone in the world)

Because it's not just about making sure the bus schedule lines up properly for everyone.
Others have mentioned that it helps when you have multiple clocks but even when you only have one clock it is important because if you are using that to measure physical phenomena, that physical phenomena is already running on the universe's clock. When you are using a timing device, there's always "two clocks" present, in a sense.
And the more precise a clock is, the more finely you can subdivide the intervals and still have them be meaningful in order to measure very fast events, or very small time differences between two events.
A: One way to think about it is that

just use a single simple mechanical clock somewhere with a wound up spring that makes it tick, and whenever it makes a tick, treat it as a second having elapsed

is not as accurate as you think. There is no guarantee that your mechanical clock will stay as it is in 100 years.
Another example can be given with the way the meter was defined first. It was defined as the lenght of a bar made by the International Bureau of Weights and Measures (BIPM). The rod can start to change shape or will deteriorate as time goes by.
As of 2019, the most accurate definition of the second can be found in this wikipedia

"The second, symbol $s$, is the SI unit of time. It is defined by taking the fixed numerical value of the caesium frequency $\Delta v_{Cs}$, the unperturbed ground-state hyperfine transition frequency of the caesium $133$ atom, to be $9192631770$ when expressed in the unit Hz, which is equal to $s^{−1}$."

A: "Why don't we just use a single simple mechanical clock somewhere..."
This is a very nice idea! And indeed for many years it was the solution to the problem of length and mass. There is (still I think) a small building in Sèvres keeping a rod of 1m length and a weight of 1Kgr.
But alas, these were the prototypes and anyone else could craft only copies. By redefining second (and meter, and Kgr and some other things) anyone now can craft the original "rod" or "Kgr" or "second" for that matter. A Cesium clock realizes exactly the new definition of the second, and can be crafted anywhere in the universe provided some homogeneity and isotropy and other conditions, thus allowing synchronization. Deep down its a matter of democracy (again). :)
A: The trick is that if you did that with a mechanical clock, what would happen is that the length of its "second" would progressively change over time as parts wear and the like. Hence you cannot know whether a "second" just ticked off really was as long or as short as one ticked off some time prior. Given you have nothing else to define a second, there's no way you could repair or rebuild the clock to fix this as you have no point of reference.
It's the same reason that the physical kilogram mass standard was gotten rid of recently. The lump of metal was really durable and well cared for ... just not durable enough and its mass slowly changed by a few micrograms.
An atomic frequency standard, though, is based on physical processes that do not change as far as we know. Even if the atomic clock gets buggy and starts reporting inaccurate time as its electronics wear out, you can just repair or replace it and compare again to this underlying standard, because it is not an artifact at all.
A: Having a central clock system has a lot of drawbacks:

*

*broadcasting means the signal takes time, so if I need a clock, it will always be somewhat behind. This cannot be fixed to the precision necessary for many modern applications

*if the clock fails, then what? Do I replace it with a new one? But will that be precise enough? Because if the new clock ticks differently, all my timings will be off. Maybe my GPS location depends on me being able to measure millisecond precision. Changing that definition changes the mathematics. So you need to synchronize clocks and make sure they don't change the definition of "a second" and that's exactly what atomic clocks are good at. That's what it means to have two atomic clocks that will only differ by a tiny amount of time after x years: They are extremely precise!

*what if I can't get the signal (ever used a cellphone?)? Then I need my own clock for the time I don't get a signal and if it's not accurate enough, maybe my application fails

*there really are applications that need to measure extremely small fractions of a second. So just getting each second approximately equal length just doesn't cut it. But you can't broadcast at arbitrary frequency, so how do I subdivide my second enough? I can do it with a local clock, I just can't do it with a radio clock...

And now for the biggest physical reason:

*

*the way the clock ticks depends on the reference frame of the clock. If I put the clock on a moving train, it will be slower to all those watching the train (special relativity). If I move it to the top of a mountain, it will go a tiny bit faster (general relativity). This means that the definition of time itself depends on the frame of reference and an "absolute time" does not exist. Therefore, we need to pick a definition which is independent of the frame of reference and use it to measure time within each frame of reference, so we NEED to have several clocks.

A: 
if the definition of the second itself is a function of the number of ticks the clock makes.

Well... the original definition of a second was a human heart beat.
In the early Roman Empire daytime was always divided in 12 hours and the nighttime was also always divided 12 hours -- for a grand total of 24 hours -- but those 24 hours were not uniform. For example, in winter "daytime seconds" were shorter while "nighttime seconds" were longer; the opposite happended in summer. What a mess.
Nowadays, a second is defined as 1/86400th of a day of 1820. And, of course 200 years later the earth doesn't rotate at the same speed anymore, so a day in 2022 has more than 86400 seconds... what produces the concept of "leap seconds". No one likes leap seconds now. I mean, who wants to count to 61 [seconds] to celebrate happy new year [every other year]?
A: Before there was GPS and before Usain Bolt took his first steps, there were ships at sea.  It's easy to tell latitude with an instrument like a Sextant and some trigonometry.  However measuring longitude without an accurate clock is very hard.
Read up on John Harrison and the development of the Marine Chronometer (start here: https://en.wikipedia.org/wiki/Marine_chronometer) to find out the importance of accurate time measurement in telling a ship where it is in the ocean.
The measurement of time and the measurement of "where you are" have been bound together through the centuries.
A: Not the accurate clock is the important fact only - more important is the comparability of their measured time-intervals. They have to be as accurate and unique as possible for comparing f. i. equal procedures all over the world.
Reliable results can be reached by same technical types of clocks.
A: The problem with a mechanical clock isn't that it's "inaccurate" relative to some other clock. The problem is that it does not measure time at the same rate: a mechanical definition of a second drifts a lot with temperature and component aging, no matter how you'd implement it, and is also changing with gravitational acceleration and with Earth's magnetic field, unless you make the entire mechanical clock from insulators. Once you get to precise time measurements, even second- or third-order effects become significant.
Both Earth's magnetic field strength and local gravitational acceleration locally drift at rates that we can now currently measure quite well, and we can do that because we can measure time without a clock affected by such effects.
Now you ask: but why we care, if we just broadcast this changing second?
Ahem. F = m*a. If the definition of the second drifts around, so would have the definition of a kilogram, in order to get accurate results from basic things like Newton's 3rd law. 3rd law is indirectly used in all kinds of measurements, so a drifting second standard would be bad news. And that's just an elementary example. All basic physical constants are inter-related to the definition of a second...
And the solution we found was finding more stable mechanical systems. It just so happens that the more stable a clock's rate, the more reproducible the design becomes, as well.
And the atomic clocks are... mechanical clocks. Quantum mechanical, but still. It's comparatively simple to replicate them everywhere you are, just given their description, since their definition is tied to... wait for it... the definition of natural numbers and counting. As long as you can agree with someone about how you count the contents of atomic nuclei, you can agree about what elements to use for the atomic clock. You can similarly count the orbitals and zero-in on the state transitions used as the time base in the atomic clock. This gets more into the Contact (the book/movie) and the way it presented the concepts of communicating basic science across civilizational divides. You have to start somewhere, and natural numbers work quite well for that purpose.
Another important thing is that time is the quantity (of sorts) we can measure most accurately. It thus helps if we can tie, at the fundamental level, the definitions of other physical constants and quantities to the definition of a second. For example, the Josephson effect ties time (frequency) to voltage, and we suddenly could improve the accuracy of our Volt standard by order of magnitude compared to previous standards of e.g. electrochemical or themoelectric nature. This circles back to Newtons, since we can relate force measurements to electromagnetic force when a certain current flows through conductors, and we can define current in terms of passage of time and natural numbers (counting the electrons!).
So, in practice it turns out that having a highly locally reproducible standard of time can be used to propagate, or disseminate, other physical standards, since they don't need to be based on any broadcasts other than static shared knowledge and definitions. That's important, as e.g. how much you pay for electricity is tied to the definitions of Volt and Ampere, and if you want to meaningfully compare electricity prices between two countries, they better agreed on how a Volt and Ampere relate to other physical constants. And such agreement is best when anyone can locally derive the needed standards without asking anyone else for anything but information.
In a nutshell, relating other physical constants to a definition of a second that is quantitatively tied to the natural numbers and fundamental physical processes allows anyone and everyone to synthesize their own physical unit and constant standards independently, without any dynamic broadcasts or sharing of artifacts.
A: Another point: you'll need synchronized clocks in many places (as argued in many other answers), and clocks that are by their nature very accurate are much easier to synchronize. And protocols like NTP (network time protocol) are not precise enough for many applications.
A: I just wanted to expand on the historical context of clocks and timekeeping.
This is an excerpt from the excellent Joel Mokyr book on technology The Lever of Riches (emphasis mine):

... [In the 15th century] The advances in clockmaking made the miniaturization of clocks feasible, and led to the democratization of time measurement.


"The clock, not the steam machine," writes Mumford with some exaggeration "is the key machine of the modern industrial age." It is mechanical, automatic, and demands a high level of precision in design and maintenance and thus served as an example for all other machinery. It created order and organization and a shared set of objective information. By the middle of the fourteenth century, the custom of dividing the hour into 60 minutes of 60 seconds each had become standard. Four o'clock was four o'clock for all individuals, an hour was an hour. This communicability of facts and concepts, the "1-see-what-you-see" stage of information diffusion, was an important element in the diffusion of innovations.


Moreover, it permitted a more accurate measurement of productivity. After all, implicit in our notion of efficiency is the need to measure time: productivity is a flow concept. Clocks brought home differences in efficiency: more productive workers and better implements and tools could he seen to produce more output per hour. Productivity comparisons became easier, and with them the choice between the faster and the slower. Clockmakers brought new standards of accuracy and complexity to the construction of mechanical contrivances, and many played important roles in subsequent inventions in other industries.

Put another way, accurate clocks were the means to an end of scientific and industrial progress:

*

*the ability to travel and keep time - that is, to set two clocks in Genoa, sail one halfway around the world and back, and have them remain in sync, no mean feat in 1492


*the ability to measure with precision the productivity of other people and processes, a key prerequisite for the mechanization and systematic organization of labor, guilds, mills, factories, large-scale agriculture


*the ability to precisely measure (and reproduce) experiments, a prerequisite to the rise of the scientific method and the "scientist", Coperncius, Brahe, Kepler, Napier, Galileo.
Max Veblen famously said, "Invention is everywhere the mother of necessity." Put another way, to answer your question: there is no reason at all a clock needs to be accurate. But the capabilities an accurate clock unlocked justified the efforts to continually produce more accurate clocks in a virtuous cycle of innovation.
