There is no difference in meaning of the two ‘present perfect’ forms (which are actually past tenses of those auxiliaries). In simple tenses, they can also be synonymous, or ‘might’ can be the past of ‘may’ (as in reported speech).
If I remember correctly, ‘may’ and ‘might’ are different in their levels of possibility. That is ‘may’ is slightly more sure (50% chance) while ‘might’ expresses more doubt (20-30% chance).
I’m afraid that is not really true, Atomos. That is one of those prescriptions cooked up for convenience’ sake. In reality, native use depends more on individual habit.