'May' and 'might' are things called 'modal auxiliaries' which is a poncy term for 'changey words'. What they do is make slight changes to the meaning of other words (the poncy term is 'grammatical mood'). It's all the 'little' words in English that allow us to tweak the intricate subtelties of a sentence.
With these kinds of words (as with all 'little' words) their use is subtle, blurred and has, over centuries, got muddled.
'Might' is sort of the past-tense of 'may' but, the two are pretty much interchangeable and tend to get used in all tenses.
'Might' is a little more uncertain than 'may' and 'may' carries the implication of permission.
To me, because 'may' gets used to ask for permission, it is more passive and puts the decision as to whether something will or won't take place outside of the control of the person saying it. e.g.
"I might go to the pub".... then again
I might not.
"I may go to the pub"... if
you are going.
Most of the differences in use are just down to modern tastes:
"Ck me" - kids today.
"Give me cake." - kids 5 years ago
"Can I have some cake?" - kids 10 years ago
"May I have some cake?" - kids 20 years ago
"Might I have some cake?" - kids 40 years ago
"Might one partake in cake?" - kids 80 years ago
"I don't want to go up that chimney" - kids 160 years ago