There really isn't any. Some people like to use the term 'Astrophysics' if they want to emphasize that they deal with the 'physics' of why things work the way they do. This is very common among the theoretical types who deal constantly with the subtleties of radiation theory, nuclear physics, stellar evolution, cosmology and the like. The older generations I think prefer 'astronomer' if their work is primarily observational, emphasizing data reduction, using telescopes to gather data, measuring star positions, brightness or color and the like. The distinction, today, between these two is so minor as not to be of concern to anyone outside the profession. Colleges may opt for using 'astrophysics' for any number of reasons including the impressive sound of 'astrophysics', or the need to show administrators that their program is just an extension of their physics department. Harvard and UC Berkeley have Departments of Astronomy, not Astrophysics.