I hear and read both the terms "aerobatic" and "acrobatic" used to describe the more extreme maneuvers such as spins, rolls, loops and more. My questions is about the terms themselves. It seems that they are interchangeable, but is one actually more correct to use than the other? Why do we have these two very similar words in the aviation vernacular that seem to mean the same thing?
Acrobatic is being incorrectly used, although it is slightly more complicated than that.
"Aerobatic" refers specifically to flight, and "acrobatic" refers specifically to feats of the human body, so referring to an aircraft as performing acrobatics is wrong.
That said, the word "aerobatic" is derived from "aero-acrobatics" and sounds nearly identical, so it's an understandable and relatively common error.
The Oxford English Dictionary's etymology for "aerobatics" is "After acrobatics" and defines it as:
Feats of expert aviation, performed esp. for display. Hence aeroˈbatic a.; ˈaerobat, one who performs aerobatics; hence as v. intr., to perform aerobatics; also trans.
So it's a new word pertaining to aviation based on one for (also Oxford English Dictionary):
A rope-dancer; a performer of daring gymnastic feats and evolutions; a tumbler. lit. and fig.
Edit: Notice the fig. at the end of the definition (which is an abbreviation for figuratively). So "acrobatic" can correctly be used to describe a person, cat, airplane, etc... performing "acrobatics".
Some of the other answers seem to claim that acrobatics would be incorrect when used to describe aerobatics, however when you look up various definitions you will find out that
acrobatics : difficult and dangerous acts done by an acrobat; also : difficult or dangerous acts, movements, etc., done by another kind of performer
So, acrobatics is just the generic word, whilst aerobatics is the more specific word describing a very specific form of acrobatics: acrobatics in the air flowing from the following etymology:
aer- + acrobatics => aerobatics
So in conclusion it's fine to use either term, especially when it's already clear that the context is about aviation. In an undefined context it might however be useful to have two words, for example
I went to an aerobatics display
I went to an acrobatics display
of course hold two very different meanings. Point is just: If you can pick, pick aerobatics, but don't go correcting people who aren't saying anything wrong.