They're very close in meaning, but "might" suggests a smaller possibility. It may sound better = it's quite possible, perhaps someone would give the OK. It might sound better = hmmm not so sure, maybe we should discuss this? In the past, "might" used to be the past tense of "may" - which is why you'll more often see "might have" instead of "may have".

Both of these are correct in any circumstance. It might sound better is less formal, and something I would say before saying it may sound better. It really depends on the situation, is there a certain sentence or phrase you are curious about? In the end either one is correct English, just depends on the formality of what you are trying to say.

They sound different. But neither is better. They're almost the same. I recommend using "might" though, because you will find it useful when doing advanced grammar.

It might sound better to say it may sound better. :P