I previously talked about a thought experiment which would supposedly show that "expansion of the universe" is fundamentally different from just "everything moving away from each other". However, discussion at the Physics Forum has convinced me that light would not be redshifted over time in a perfectly reflecting mirrored box despite the light being stretched along with the expansion of space.
Moving walls in comoving coordinates
One way to look at the problem is to realize that in comoving coordinates, the walls of the mirrored box are actually moving towards each other. Comoving coordinates is the coordinate system where the universe is expanding the same way everywhere and in which objects staying "still" observe light being redshifted, and in this coordinate system the walls of the box are actually moving towards each other and causes blueshift in the light, which cancels out the redshift due to the expansion of the universe.
This can be confusing, but then again talking about things moving and staying still can be confusing in relativity with all the different reference frames and coordinate systems that could be used. And a qualitative argument like this doesn't tell us just how much the blueshift cancels the redshift. But it does illustrate what happens in the "space is expanding" interpretation.
Moving away as an approximation
Another way to look at the problem is to use the fact that for short distances, everything simply moving away from each other is a very good approximation to the expansion of the universe. This might be in conflict with what I said about space expanding being fundamentally different from things just moving away, but it's not. You just need to remember that it's only a good approximation for short distances and that it's a very bad approximation for very large distances. If you wonder how distant galaxies could move faster than the speed of light, then you're forgetting this.
In any case, thinking about the mirrored box in these terms makes it clear that there should be no redshift since the walls are not moving at all from this perspective. I do not know if there would be a very small shift in wavelength over billions of years from errors in the approximation, but if there is any, it would be nowhere near the redshift seen in the cosmological microwave background.
It would be nice to be able to show that "expansion of the universe" is different than "everything just moving away from each other" with an obvious difference, and my thought experiment with the perfectly reflecting mirrored box tried to do just that. It unfortunately does not work out, so one will still have to deal with the subtleties of reference frames, coordinate systems, definitions of distances and speed, etc. to understand how the expansion of the universe is not a just stuff moving away from each other uniformly in a fixed and flat space-time.