Is there any physical evidence for motion? Let's say that we have 2 tennis balls in space, one being in motion (say, pushed by an astronaut), and the other one still.
If we could take a snapshot of both tennis balls, would there be any evidence that could suggest that one is moving and the other one is still? Is there anything happening, at the atomic level or bigger, being responsible for the motion?
If there isn't, and both balls are absolutely identical, then how come one is still and the other one moving? Where does the difference of motion come from?
 A: According to classical physics: no. It is impossible to tell how fast something is moving from a snapshot.
According to special relativity: yes. If we choose a frame of reference where one of the balls is at rest then only that ball will look normal. The other ball is moving in this frame so it will be length contracted. If its rest length is $L$ then its length will now be $L\sqrt{1-v^2/c^2}$. Since $1-v^2/c^2<1$ the ball will be shorter in the direction it is moving. 
According to quantum mechanics: yes? In quantum mechanics particles are described by a wavefunction $\psi(x)$ which (handwavingly) says  how much of the particle is present at a certain point. A tennis ball is also described by a wavefunction which you can get by combining all the wavefunctions of its atoms. The wavefunction actually contains all the information you can possibly know about an object, including its velocity. So if you could pause time and look at the wavefunction you would have enough information to know its (most likely) velocity. In real life you can't actually look at wavefunctions: you have to perform an experiment to extract information from the wavefunction. At this point you might wonder if that still counts as taking a snapshot.
A: 
If we could take a snapshot of both tennis balls, would there be any evidence that could suggest that one is moving and the other one is still?

We can't.  Problem solved.
Well, almost problem solved.  So in reality, we can take shorter and shorter exposures.  I can take a 1 second exposure of the scene, where the moving tennis ball will be heavily blurred while the stationary one will be crisp.  I can capture the same scene at 1/100th of a second, and moving ball will look more crisp like the stationary one.   I can capture the same scene at 1/1000th of a second, and it will be very difficult for the human eye to discern which one is in motion.  I can make these snapshots shorter and shorter. Indeed, we have looked at imaging scenes at such exacting shutter speeds that we can watch light propagate through the scene.  But we never quite hit a perfect standstill.  We never hit an infinitely fast shutter speed.
Now forgive me if I handwave a bit, but there is an unimaginably large body of evidence that motion exists.  In particular, you'll fail to predict very much if you assume no motion occurs.  So from that empirical point of view, we should find that motion exists.  From a philosophical point of view, there's some interesting questions to be had regarding endurable versus perdurable views of the universe, but from a scientific perspective, we generally agree that motion exists.
So how do we resolve the conundrum you are considering?  The answer is calculus.  Roughly 400 years ago, Isaac Newton and Gottfried Leibniz independently developed a consistent way of dealing with infinitesimally small values.  We generally accept this as the "correct" way of handling them.  It does not permit us to consider a shutter speed which is truly infinite, letting us isolate a moment perfectly, to see if there is motion or not, but it does let us answer the question "what happens if we crank the shutter speed up?  What if we go 1/100th of a second, 1/1000th, 1/100000th, 1/0000000000th of a second and just keep going?"  What happens if we have an infinitesimally small exposure period in our camera?
Using that rigor, what we find is that modeling the world around us really requires two things.  The first is the values you are familiar with, such as position.  And the second is the derivatives of those familiar things, such as velocity.  These are the results of applying the calculus to the former group.
We find that models such as Lagrangian and Hamiltonian models of systems work remarkably well for predicting virtually all systems.  These systems explicitly include this concept of a derivative in them, this idea of an "instantaneous rate of change."  So we say there is motion, because it seems unimaginably difficult to believe that these patterns work so well if there was not motion!
As a side note, you set up your experiment in space, so there's nothing much to interact with.  However, had you set the experiment up in the water, you would find the chaotic flow behind the moving ball very interesting.  It would be ripe with fascinating and beautiful twirls that are very hard to explain unless associated with some forward motion.
A: Your question assumes one ball is moving and the other is still. That assumption is meaningless without specifying a frame of reference. All motion is relative. To each of the balls it would appear that the other was moving. The 'evidence' that they are moving includes the fact that they would appear smaller to each other, and that their separation was changing. 
A: You are limiting your snapshot to a 3D picture.
If you took a 2D snapshot, it would be impossible to tell how deep your tennis "balls" are (in addition to being unable to tell their motion).
So, take a 4D "snapshot", and all'll be fine.
A: 
If we could take a snapshot of both tennis balls, would there be any evidence that could suggest that one is moving and the other one is still? Is there anything happening, at the atomic level or bigger, being responsible for the motion?

If the balls are truly identical and you are at rest with respect to one of them, the light of the one moving will look more red or blue, depending on whether it is moving toward or away from you, by the Doppler shift.  This would be most evident if you were positioned between the balls and on their axis, but you would always be able to do it as long as the moving ball is at least partially approaching or moving away from you.
A: Cylinders Don't Exist
If I show you a picture of two round objects and tell you that one is a sphere and the other is a cylinder you are looking at head-on, how can you tell whether I am telling the truth or lying?  You can't, and therefore, I conclude that there is no difference between spheres and cylinders, because we lack the proper evidence for their existence.
Projection
The point here is that motion requires time, and a snapshot is a projection of a 4-D extended object into 3- or 2-D.  The most naive such projections will necessarily destroy information about additional dimensions.  If I remove one of the axes which would help you distinguish a cylinder from a sphere (ignoring light reflections, etc.), this is no different than you removing the time dimension to make it impossible to distinguish between a moving or a static object.
Conclusion
If you want to establish the separate existence of spheres and cylinders, you must examine them in all the dimensions which make them different.  If you want to establish the existence of dynamic 4-D objects (objects which vary in the time dimension), you must examine them in all the dimensions which differentiate them from purely static objects (ones which are constant along the time dimension).
A: It is about the frame of reference, in the frame of reference of the tennis ball pushed by the astronaut, it could be considered as standing still and the other ball, the astronaut, and everything else as moving. For the frame of reference of the other ball it could be considered as standing still, and the first ball as moving. If you were with either one, in it's frame of reference, all of the physical laws of the universe would be the same and neither could be preferred as absolute. This is one of the basics of relativity.
A: The photos would look identical, but you would have to take each photo from a different inertial frame of reference. You have to be moving in a different speed in a different direction to take the photo. This shows that there is inherent differences between objects in motion.
A: 
If there isn't, and both balls are absolutely identical, then how come one is still and the other one moving? Where does the difference of motion come from?

I don't think this question is nearly as perplexing as you might think nor do I think it requires sophisticated physics like the best answer describes. Ask yourself, how do you show with a snapshot that a bowl of soup is at a cold 5 C vs a warm 45 C? Or how could you show that a radio is turned off or is blaring music? Intuitive solutions to these questions would be to take a picture with a thermometer or an oscilloscope attached to a microphone respectively in the same frame. 
The easiest way to show with a snapshot that a tennis ball is moving, is by taking a picture with a speedometer reading in the same frame as the ball. 
These examples are hard to show directly in a single snapshot in time because they all involve the collective motion of small particles (uniform velocity for motion, random for thermal, and periodic for sound). And motion is described as the change of movement with time, but a snapshot captures an instance in time not a change.
A: It is perhaps interesting to think about Mach's paradox in this context.  I'll get back to your question and the limits of the two-body way of discussing special relativity in the end.  One form of the paradox is this: imagine a bucket of water standing on the floor.  The surface is (almost) flat.  Now start spinning it.  The water's surface begins to form a paraboloid.  How does the water know it's spinning?  Why is the frame of reference in which the water's surface is flat the same frame in which the stars do not move relative to the bucket (which almost coïncides with the frame where the Earth is still)?
The answer is that the presence of the stars determines the global geometry of the universe, and thus the local free-falling frame in which the bucket finds itself up to small corrections due to Earth's gravity and rotation (which is in free fall around the sun which is in free fall through the galaxy which is in free fall through the universe).
Now how does all this relate to your question?  Well, we can determine accelerations relative to a global frame given by the fixed stars with as simple a tool as the aforementioned bucket.  But if we accept that global frame as special, then we can also detect motion relative to that global frame, which is in a sense absolute as it is given by the universe in its entirety.  To do this, you'd need a long exposure and a clear night-sky.  You would then compare the motion of your tennis balls to the motion of the stars and you could in a meaningful sense call the difference of the motion relative to the stars an absolute motion, as it is relative to the universe as a whole (well, to good approximation depending on how many stars you can actually photograph).  Since this is literally the opposite of what you had asked, it doesn't literally answer your question, but I think it answers the same question in spirit, namely whether there is a physical difference between "moving" and "stationary."
NB It may seem that I'm upending all of special relativity by that line of thinking but that's not true.  Special relativity is a good law of nature, one just has to be aware of other objects which are present when applying it, and whether they have any influence on the question studied -- and that statement is a trivial truth which certainly was on Einstein's mind when he wrote the time dilation law for the first time.
A: It is possible to measure absolute motion relative to the cosmic microwave background.. A system in rest with the moving ball would measure a dipole in the background radiation. 
A: If we look past your example with snapshots we can just look into modern technology and find a little thing called videos. They can record motion pretty easily. "Is there any physical evidence for motion", videos?
