Except when we use right to denote direction, what is the difference between these two terms?
Also, which one is the preferred construction between these two
Am I right?
Am I correct?
I'd say that there is a subtle distinction between the two, in that "correct" implies that the given answer (or answers) is definitive and absolute, as might be the case, for example, with a simple maths problem.
In contrast, "right" would be better for an answer which, although considered appropriate, is still matter of opinion (even if most people agree). This might apply, for example, to a question of ethics ("is this the right thing to do?"); again, using "correct" here would imply that there is an absolute answer that no-one could possibly disagree with.
"Right" can still be used in places where "correct" would be appropriate (such as a maths problem), but not the necessarily the other way around.
They are synonyms, but I'd reserve the use of correct for either formal usage ("The mathematical proof was shown to be correct") or as a verb ("correct one's behavior")
The writer will otherwise risk sounding pedantic or pretentious.
I always make sure to use 'correct' when someone is asking about directions. Otherwise it's like Laurel and Hardy:
Them: So I go left on Elm Street?
Them: Oh, I'm sorry. So I turn RIGHT on Elm. Then-
Me: No, no, you go LEFT on Elm.