The nuance difference between 안 좋아요 and 나빠요 is the same as the one between saying someting is "not good" and "bad" in English :)

Actually they are not that different. 안 is used to make negatives so the negative of ‘feeling good 기분이 좋아요’ (which is ‘not feeling good 기분이 안 좋아요’) has a similar meaning with ‘feeling bad 기분이 나빠요 ’ (rather than ‘feel nothing’...!!) Umm.. but there is a small difference in that 기분이 안 좋아요 means feel bad and 기분 나빠요 means feel so bad or upset..? It means that 기분 나빠요 has a stronger meaning! (And I agree to the very answer that the other one wrote!)