It depends a lot on tone and body language, but "right" usually means "I agree with you" while "correct" can come off as them deigning to give their approval fr

It depends a lot on tone and body language, but "right" usually means "I agree with you" while "correct" can come off as them deigning to give their approval from a high place, like "I'm the one who actually has the knowledge, but you happen to be correct in this specific instance". But it's more about the person and their tone, the people who feel strongly about this are probably who have experienced it from someone insufferable who uses it as a power play. I.e. if you're not intentionally using it this way, you don't have much to worry about.