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You can't, for the same reason that you can't tell the difference between acceleration due to gravity when you're sitting in a chair at home, and acceleration from a rocket, when you're sitting in the pilot's seat. This is known as the "equivalence principle", and there are different formulations of it. It's a big topic.

This is what bothers me. If they are equivalent, then how can we say the expansion of the universe is due to space expanding and not movement within space, unless we can distinguish between the two?

Because, first of all, the universe is the same in every direction we look (which tells us that if the universe exploded outward from a central point into empty space we'd have to have ended up extremely close to the center of the matter distribution), and second of all, because of the cosmic microwave background radiation (which tells us that the universe was dense to the point of opacity in the distant past, in any direction we can look).

This is what bothers me. If they are equivalent, then how can we say the expansion of the universe is due to space expanding and not movement within space, unless we can distinguish between the two?

Depends on the distances you want to talk about. Luckily, the expansion of the universe is nearly immeasurable at close distances, so you can throw it out for most measurements, like measuring velocities within the galaxy. But once you get outside what is assumed to be the galactic halo (about 300 kilopasecs, or about 1200000 light years), then you have to factor it out. Luckily, the expansion of the universe has been occurring nearly linearly in recent times, so you can just measure the velocity, and since the Hubble velocity of expansion is generally larger than the velocity of motion through space, you can fit that distance with other distance estimates, like Cepheids, Type I supernovae, and other methods. Whatever is left over is a good estimate of the space velocity of the source along the line of sight.

There are other caveats, but that should give you a good idea of how it is done.