I have these distinctions in my mind:
Astronomy: A science concerned with discovering and cataloguing celestial bodies. To map the Universe, if you will.
Cosmology: A science concerned with the history of the Universe. How it developed over time and is developing today. How galaxies and other systems came to be and where they are going.
Astrophysics: A science concerned with the "behavior" of celestial bodies, identifying what forces they exert and are subject to, as well as figuring out what happens at the atomic level in stars, pulsars, black holes etc. It gives Cosmology the tools it requires to make its discoveries and explanations.
Astrology: A ridiculous pseudoscience that has no logical grounding whatsoever, and should not be studied by anyone sane of mind.
In this way, Neil Degrasse Tyson and Stephen Hawking would be cosmologists and astrophysicists, but not astronomers. I don't think they care about classifying recently discovered galaxies!