Which is stronger, a rope without knots or a rope with knots? Suppose I have two identical ropes, one of which I manually tie some knots in. If I use them to hang clothes, which one is more durable? 


Personally I think that rope with knots will be more durable, but I can't come up with a satisfactory reason.
 A: A curvature in the rope causes the tensile strength of a rope to decrease. The strongest configuration is the one where the stress distribution across the rope cross-section is close to uniform, i.e. when the rope is straight.

When the rope has a knot it may seem more stable, simply because the overall rope is thicker. However the rope will usually not break in the knot, but at the end of one, where there is a curvature.
If you are looking for data on the topic there are two main applications: industrial ropes and steel wires and slacklining. For the former knots are not very common (try tying one in a steel rope...), so I will briefly talk about the latter as an application of the concept.
One place where one has to tie a knot is the ends of a slackline. In fact there are special devices, called slackline weblocks that try to achieve that with minimal curvature (see picture). 
If you are searching for actual data on this the best place to search might be the Slackscience Blog by Balance Community. They have tests of different weblocks and how investigate how their structure effects the tensile strength.
A: Every rope is weakened by tying a knot!
The physics behind this phenomenon are very complex (I for one, do not know them, or whether they have been understood completely), but it has been empirically shown repeatably that any knot weakens the rope by about 30% (from maximum load when no knots are present), and when the rope breaks, it does so at the knot, or its vicinity. 
For reference (one out of many sources that would support the claim):
"Knot Break Strength vs Rope Break Strength", The National Speleological Society
A: All the answers so far have talked about strength, but the OP asked about durability.  If the clothes were pinned onto the knots only, then it should be more durable than an unknotted rope as the wear would be concentrated on an area with multiple layers of the rope.  The load on each part of a knot is less than on an unknotted section, so any fraying on the knot would not affect the strength over time as much as if the unknotted rope were frayed.
Ultimate breaking strength is weakened by knots as has already been stated multiple times.
http://www.paci.com.au/downloads_public/knots/03_Cordage_Institute_Tests.pdf
A: Knots are weak points in tensile strength, simply because they create extra and un-necessary bending stress and twisting the strands beyond their normal service load.
The geometry and mechanics of knots creates sharp bends around small radios of curvature which is promoting failure and cutting of strands and fraying of the material.
When you load a rope with tension it reacts by stretching, and when you remove the load it shrinks back to its original size like a spring. However in the confines of the area of knot it wont stretch so you create concentration of stress on those points on rope immediately before and after the knot, which soon promotes start of fatigue cracks.  
Not only knots are weak points but even after untying them if they have already formed kinks in the rope it is a weak point. As we all intuitively check any rope for its being straight and not kinked before using it!
A: I will assume your question concerns the force you can apply to the ends of a rope without it breaking, and how this force depends on the rope being straight or knotted.
While some of the other answers deal with the details of the rope (e.g. how it is built from fibers, and how knots imply curvature, bending stresses and such), let me propose a more fundamental approach, based on symmetry, to explain why the knotted rope will not tolerate more tension that the straight rope.
Let us assume that the unknotted rope is ideal, in the sense that it has the same properties everywhere along its arc. This gives the rope a certain symmetry, since it does not matter which point along its arc we discuss; every one of them is the same.
Now, let us introduce a knot anywhere on the rope, or more than one if you like. This breaks the symmetry, since now, there are clearly points at the rope (the knots and possibly their surroundings) which differ from the straight rope segments. What we will not allow, though, is "infinitely many" knots, so that every segment of the rope is now "knotted". (What "knotted" is supposed to mean exactly will not be our concern.)
Then, there are three possible outcomes:


*

*For every segment of the knotted rope, the "strength" (i.e. the tension it can bear) remains unchanged. Then, the rope's strength stays the same.

*For every segment of the knotted rope, the strength is reduced, compared to the un-knotted rope. Clearly, the strength has decreased.

*There are segments of the rope with unchanged strength, and some with changed strength.


To analyze the last case, let us introduce an intuitive assumption about physics: when the rope is under tension, it will break at its weakest point.
So, if the knots introduce segments of inferior strength into the rope, the weakest of those segments will break before the segments that have the original strength will fail. The rope as a whole is thus weaker.
On the other hand, if the knots only cause segments of the rope to get stronger, then the rope will break at any of the unknotted segments, giving the knotted rope the same strength as the un-knotted one.
Obviously, if you have a long rope, you may make it stronger by folding it in half, and perhabs twisting the two strands in some way. However, this would count as "infinitely many knots", which we ruled out above.
A: The fibers in a rope strand are 'layed' stretched out straight, then
twisted.  Multiple strands are combined, by twisting, to make the rope.
When a rope is pulled taut, all the fibers have similar stress, and all contribute nearly equally to the tensile strength.  When in a knot, however, the fibers on the inside of the curve are not in tension, and those on the outside are stretched as they follow the curve.  Strain is uneven.
Under tension, the rope is stronger if there are no knots, because the uneven strains caused by the knot shape can snap the most-stressed fibers at tension levels that do not threaten to damage a straight rope.  Various knots, and
types of splices, have been tested to see how well they preserve the strength of
the original line; an eye splice with a thimble (steel internal support) is one
of the better designs; the overhand knot (illustrated in the question) is one of the worst. 
A: The answers above are already very complete considering strength. There is however one other point that I'd like to add.
If you have a knot in a rope that gets wet, the knot will stay wet longer than the rest of the rope. This will allow mold to grow more easily in the knot, where it has a wet (and probably warm) place to stay and grow. Of course it goes without saying that mold will weaken your rope. Side remark: you don't need knots to grow mold; in time even a straight rope will probably start growing mold.
Another point I'd like to make is that for the application you are mentioning it probably will not matter since the weight of your clothes is not that large compared to the breaking strength of rope. Of course if you really put a lot of tension on your system this can start to become relevant.
Finally the rope will probably be hanging permanently outside exposing it to UV and hence UV damage on your rope.

Conclusion: If you'd like to put knots in your rope you can do this for a regular clothesline. Your biggest enemies will probably still be moisture and UV damage, which will cause the rope to decay over time. The system with knots will probably not last quite as long as the system without knots. But from practical experience I knot that the difference in lifetime will be small.

A: If you have a damaged (usually frayed) section of rope then it is possible to knot in another section of rope to take the load.  Typically you would do this with a sheepshank or one of its variants.  In this case the knotted rope will be stronger than before, because its strength was seriously limited by the damaged section. Now the damaged section is no longer taking the strain, you've fixed that weakest link.
However if you started with two identical ropes, and then damage-and-sheepshank one of them, the undamaged/unsheepshanked rope will be stronger.  The sheepshank fix is stronger than a damaged rope; but the sheepshank itself degrades the rope's load-bearing abilities (see answers above), so the undamaged/unsheepshanked rope is strongest.
A: https://en.wikipedia.org/wiki/Knot#Strength
Knot strength is studied fairly extensively with fishing knots in search of the "holy grail" of knots, a 100% strength knot, one which is not the weak link in your line.  With mono-filament fishing line, there are a number of issues which cause the knot to be weaker than the line.  "Burn" occurs when tightening a knot, where friction causes the mono to heat up and even melt and reform.  With heavier gauge of twisted rope this may be less, but is still real.
The uneven tension in the knot will cause distortion and stress to not be evenly distributed, causing uneven wear to the line or rope.
Most knots have some level of slippage when stress is applied.  The slippage will cause the rope or line to rub while under stress and slowly cut, weakening the line.
In fishing knots, the general goal to having a stronger knot is to lubricate the line while tightening the knot to minimize the friction/heat and tie a knot that locks in place to minimize movement.  With this it is found that each crossing which is made, that is line over line, reduces line strength by about 10%, so with fishing line 90% strength of a knotted line as compared to no knots is about the best that can be made.
Some of this varies by diameter, with different configurations favoring different knots due to line slippage, so fishing knots that are relatively strong might not be good choices for braided lines.  That said, the general principle remains that a knotted line will be weaker than one with no knots.  An exception might be forming a compound rope by knotting more than one line together in parallel which could be stronger than a single line but more stable than multiple lines simply looped together.
