I Can't, I Don't: 안 vs 못

Two forms of negation, both placed immediately before a verb or adjective:

안 — you didn’t do it, but you had the ability to

못 — you couldn’t do it because you don’t have the skills or ability

우유를 못 마셔요.

I can’t drink milk. (I’m lactose intolerant)

우유를 안 마셔요.

I don’t drink milk. (I think it tastes awful)

수줍어니까 자주 한국말을 안 해요.

I don’t speak Korean often because I’m shy.

한국말을 배워 본 적 없으니까 한국말을 못 해요.

I can’t speak Korean because I’ve never learned it.