It's a good question. The first part of the answer is, there's no difference between who is moving, as long as you're both moving uniformly (i.e., with constant speed and direction). If you're in a spaceship speeding away from Earth, you'll see the clocks on Earth as slow, and people on Earth will see your clock on slow. Everyone is free to decide that they're the one standing still, and the other person is the one moving. It's not a paradox that both clocks appear slow; measurement of time is relative to your motion, and you can't check whose clock is "really" slow, because the two clocks are steadily getting further and further away.
If you turn your spaceship around and come back to Earth, so that you can directly compare the two clocks, the situation is no longer symmetrical. You turned around, going through a period of hard acceleration to change your velocity from nearly light speed in one direction to nearly light speed in the other direction. There are a number of ways to work out what happens in this case, but it's pretty easy to show using relativity that the clock that accelerated will always show less time elapsed than the clock that did not, once they're reunited. Moreover, all observers, regardless of how they are moving relative to the Earth or the spaceship, will agree on this.