What's the difference between mechanical and electromagnetic waves? I have a life science degree and even worked in research for a few years. So I feel I should be able to answer this question for myself, but yesterday my daughter blindsided me by asking why, if sound and light are both waves, light can travel through the vacuum of space but sounds can't?
So, I told her that one was a mechanical wave and other was an electromagnetic wave and ... then realised that was the extent of my knowledge. I was parroting stuff I'd learn years ago without any real understanding.
Can someone explain to me, ideally with science but without maths:


*

*what the difference is?

*why one requires a medium for propagation and the other doesn't?

*how come electromagnetic waves can still propagate via a medium (i.e. heat waves still carry heat energy through air, or the body)?

*whether all electromagnetic waves are quantized (i.e. light waves are made of photons but ... heat waves are too, right?)

*whether it's possible for a wave of each type to exist at the same wavelength?


EDIT: stepping back, it appears I have caused confusion born of ignorance. Essentially, while I can understand the properties of a mechanical wave simply by observing it, I am struggling to understand how intangible things like heat and light can have the same properties. 
 A: There are many more differences than similarities, so I would flip the question around and ask: "what are the similarities?".  
The only similarity is that in both mechanical and electromagnetic waves, each is as oscillation where each point of the wave can act as an independent source of a spherically propagating waves that add together (or a circular wave on water, or a linear wave in a string).  All the similarities (wavefronts, bending around corners, interference, etc), follow from this.
Everything else you ask (nature of quanta, medium, etc) involve independent detailed description of the specific wave, and are generally different based on the different physical mechanisms.
Finally, you mention "heat waves".  If by this you mean infrared radiation, that is of the same set, but if you mean heating one end of a bar getting the other end hot, that would be "diffusion", not a wave.
A: The main similarity between mechanical sound waves and electromagnetic waves is that it both corresponds to the propagation of some "disturbance", in one case it is the local increasing or decreasing of pressure ; in the other it is the excitation of the electromagnetic field.
Sound waves require a material medium to propagate because of their very definition : they correspond to a local increase in pressure, which cannot exist in the vacuum.
Electromagnetic waves are different because their means of propagation (the electromagnetic field) can exist in material media as well as in the vacuum : the physical vacuum isn't just empty space. The difference being that when it propagates inside of some medium, the characteristics of the medium (local density, polarisability, temperature...) can affect the way electromagnetic waves propagates with respect to its propagation in the vacuum.
A: First of all, a wave is something that propagates and bears some periodicity during its propagation. Mathematically, this can be elegantly described by a certain set of mathematical formulae, completely independent of the physical meaning. So, there are many similarities between, e.g. sound and light waves, but also differences that are unique to each of those physical phenomena.
What is the difference between sound and light waves and why does one require a medium 
Light/electromagnetic radiation is a form of energy of its own. Once "created", it can exist on its own until it is converted into another form of energy, e.g. when the wave is absorbed in matter. A sound wave on the other hand is a periodic motion of matter (air when you speak, water molecules when you think of the noise from ships' propellers, iron if you think of the vibrations you create by hitting an anvil with an hammer). If there were no molecules, an initial vibration would not move anything... and therefore there wouldn't be a sound wave.
Sound is a periodic density modulation of matter. Therefore it needs the matter to exist and the matter is the medium that the wave traverses. Light is a form of energy that can exist on its own. Therefore it does not need a medium to exist.
Why can electromagnetic waves travel in a medium
The answer is complicated, since electromagnetic waves cannot "just travel in any medium". You can test this by yourself: On a bright, sunny day, place your hand before your eyes and turn your face right towards the sun. Can you see through your hand? No, you can't. Therefore, the light from the sun does not travel through your hand.
Light as an electromagnetic wave can be affected by electromagnetic "objects" - and those exist plenty in any form of matter, because molecules and atoms are made up from electric charges. Those electric charges can be combined in many different ways. Some of them allow the light to pass through, some of them don't. To put things in very simple (but hopefully better understandable) terms: Typically light bounces off of molecules and around in the matter. Due to this bouncing, the light is slowed down, i.e. the velocity of light is lower in matter than in vacuum. Another thing that happens often is that light just has the right frequency and all its energy is gone after it hits an appropriate molecule. The energy is then converted into vibration and rotation of the molecule - and the light itself is gone and can therefore not pass the matter any longer. This is what happens in the little experiment proposed at the beginning of this section. If you happen to have a matter where light can just go through, then you could think of the molecules as receiving antennas which are not tuned to the correct frequency. The light wave does not "see" those antennas and just passes between them (and there is lots of space between molecules).
So, light can travel in matter only, if the matter physically allows for that. Typically, it doesn't. Light can be absorbed, reflected, or transmitted. It depends on the kind of light (i.e. frequency of the electromagnetic radiation) and the kind of matter that it interacts with.
Are all electromagnetic waves quantized?
Yes. They are. Also heat radiation. Heat is a critical topic, since the kind of heat that we humans can feel is "incoherent motion of molecules", which can be mediated by matter and radiation. The latter one is the one you were asking for.
Yes, they are.
Can waves of each type have the same wavelengths?
Yes, this is entirely possible, but some physical limits must be respected. I.e. it does not make sense to have a sound wave that has a shorter wavelengths than the size of the molecules that it its medium is made of. Sure, there are other limiting factors in the sound wave example that have to do with inertia of molecules, viscosity, etc... But let's say you have a sound wave with 440 Hz... It is no problem to have an electromagnetic wave with this frequency. You just won't see or feel it, because of its low frequency, that is well below anything we humans can perceive.
Yes, it is possible, but don't mix up the effects - you would not start to hear such a light wave, nor would you start to see the sound wave.
A: There's a really important thing going on here.
Like Tom10 said, there are more differences than similarities. For example, if you travel alongside a sound wave, everything is fine. Well, you get sonic booms, but you can still do it. If you try to travel alongside an electromagnetic wave, you run into problems with relativity. You just see the wave get redshifted more and more while it travels in the same direction at the same speed.
Compare this to other types of waves. Sound waves have a maximum frequency due to atomic details. And different modes of propagation. There are weird nonlinearities. And if the amplitude is too high, your solid will break or melt. You have $10^{23}$ particles bouncing around, and there are a lot of different phenomena!
Similarly, surface waves on water have lots of ugly properties. They can break, they have oddly shaped patterns called cnoidal waves, and of course, if you get a cup of water and pour it on the ground, there's not much wave behavior at all...
And finally, heat! I think by "heat waves" you mean precisely "infrared radiation", and by "light waves" you mean precisely "visible radiation". Regular, run-of-the-mill "heat" is basically just sound. Vibrations of atoms. (Interestingly enough, there's also a quantization that comes about for sound, and you can use the quantum mechanics of quantum sound to calculate thermal properties of matter.)
You might ask why so many different phenomena (waves on the surface of water, sound waves, electromagnetic waves, and even gravity waves) can be described by the same notion of a "wave" in some approximation. (Electromagnetic radiation is the best approximation of an ideal wave in that list, by the way!) The reason behind this is that somehow, some way, classical physics only needs to talk about accelerations. And when you deal with approximations involving accelerations (small amplitude sound waves, small amplitude surface waves on water, small amplitude gravity waves), it's very hard to wind up with anything other than a wave equation. (Related shameless self-plug: "Why are sine/cosine always used to describe oscillations?")
The real answer for your daughter, is that sound waves are an emergent property when you have $10^{XX}$ particles bouncing around, whereas electromagnetic waves are a property of space itself. You can say there is a sort of medium through which waves propagate. That medium is space. Space has properties. It has an energy density from electromagnetism. It has an energy density from gravitation. General relativity tells you energy density=mass, so space has mass too. It's a medium. Just a bit of a weird medium which obeys Lorentz invariance, and Lorentz invariance is a bit counterintuitive for our hunter-gatherer minds, so a lot of physicists/teachers prefer to tell people it's "not a medium". All they mean to say is that it's not a medium in the same way that a block of metal is a medium.
Maybe it helps to imagine this: Imagine somewhere deep in outer space. Maybe you imagine a big box a meter cubed, and the only thing inside it is a single hydrogen atom. Sound can't propagate, there's nothing for the hydrogen atom to hit to pass its energy on to. Now imagine there's a light wave passing through the box. Well, what is light? Maxwell teaches us it's just part of the electromagnetic field, and electric fields accelerate protons and electrons. If you looked closely, you'd see the proton being pulled one direction and the electron being pulled the opposite direction. The whole thing vibrates periodically. An electromagnetic wave is propagating through empty space.
If you fill the box up with hydrogen atoms, space is still there, so the electromagnetic wave can propagate just fine (although it's effected by the matter creating its own electromagnetic waves in response. Matter can react in a lot of different ways so that can be a very complicated situation).
A: There are many waves generated by the release of energy but the two major types of waves are physical and electromagnetic. Physical waves need a medium to go through, and electromagnetic can go through many substances including a vacuum. 
The term "wave" refers to both physical and electromagnetic. The different components include; the crest (top of a wave); trough (bottom of the wave); wave height (how high is the crest); and wavelength (distance from crest to crest or trough to trough). These terms even apply to the waves produced in oceans.
Waves are essentially a way in which energy can be transferred from one place to another.
A: The most important difference between water waves and EM radiation is the fact, that the first has a measurable amplitude and a wavelength. EM radiation does not have such direct measurable properties. A amplitude is not defined nor measurable. What we can measure are the properties of a radio wave.
The radio wave is a - modulated by a wave generator - stream of photons. The generator accelerate periodically electrons in the antenna rod and by this the electrons emit photons in a swelling on and off stream. Since the velocity of light is the same for photons of all energies (in air this velocity is a little bit less than in vacuum), the wavelength of this modulated radiation is simple c/f, there f is the frequency of the wave generator and c is the velocity of light (in air for radio waves on earth).
The wavelength of monochromatic light is measured indirectly by the measurement of the mirror movement in an interferometer measurement. This is a indirect measurement of the mirror shift and the overlapping of two intensity distributions from different paths of a light beam. Such a measurement presupposes that the distance of the shift of the mirror is not only proportional to the wavelength but equals the wavelength. So it is without any doubt, that the interferometer is able to set in relation the frequency of two monochromatic light beams, but to make a direct measurement of the wavelength is not possible.
But an oscillation of a light beam was never measured except the case of a beam modulation like in any bulb with the frequency of 50 Hz (or 60 Hz or another frequency of the electric generators for AC current).
For completeness, it should be mentioned tha every photon has oscillating electric and magnetic field components and this can be somehow be interpreted as a wave propagation.

A: Purpose of wave is to propagate the energy. Mechanical waves do this by 'strain' of meadium. For example, sound wave propagate by by change in pressure of medium. Thus a medium is necessary for this kind of waves.
But, an electromagnetic wave, on the contrary occure due to variations in electric and magnetic field associated with it. If you take a magnet or a charge into space, then obviously magnetic or electric field due to this extend towards infinity. So, this means that change in electric and magnetic field takeplace without the help of medium. And this is why the electromagnetic waves require no medium for propagation.  
Concludingly, the mechanical wave propagate because of properties of medium while the electromagnetic waves propagate because of change in electromagnetic field associated with it, for which a medium is not needed.  
I think this is the main problem focussed in your question, and you can ask rest of your doubt as seperate questions. 
