When I have to compare two things A and B, can I say "a comparison between A and B" or "a comparison of A and B". Which is best ?
To keep things super-duper simple:
Both are fine. They mean exactly the same thing.
Use either one. No one will misunderstand you. Both are equally common in everyday usage. In fact, if anything, I would say that "a comparison of A and B" is perhaps more common, even though it may sound slightly less correct. This Ngram would seem to back me up.
If you compare A with B, then the comparison is between A and B. You could also say "comparison of A with B" as in the following sentence.
The immediacy of effect is assessed both with the observed data from phases 1 and 2, and a comparison of the observed data in phase 2 with the projected data from phase 1.
You could also say "comparison of A and B" as in the following sentence.
Counseling smoking parents of young children: comparison of pediatricians and family physidans.
You use between when you are comparing two things and among when comparing more than two things.
Between my brother and sister, my brother is easily the most spoiled.
Tom is the smartest among my friends.