Interesting question. Actually I wanted to go into space physics, but got into a school where they do astrophysics, so I ended up doing the latter. In my experience, space physics is usually short for space plasma physics. It's basically the study of the Earth and solar system, including magnetospheres, stellar wind, the physics of the sun, magnetohydrodynamics, etc. Space physics differs from astrophysics in that in astro, we can't actually go to the sites that we're observing and take in situ data. Space physicists, on the other hand, can send up probes and take all kinds of measurements about the solar system.
Of course, based on what Astronuc said, it seems that some departments or labs use the term "space physics" to refer to what is really astrophysics.