Are "may" and "might" synonyms?
Sometimes. Sometimes not.
As BillJ tells you, may has no infinitive form, and might is historically its so-called ‘past’ form.
In some contexts might does in fact act as the past-tense form of may. If John tells you on March 1 “I may go to Paris next week”, you could report this on July 1 as
John said he might go to Paris the following week.
In this case, may and might contrast in tense. BUT:
It is also true that John might (!) have said, in exactly the same context and with exactly the same meaning, “I might go to Paris next week”.
In this case there would be no change to the verb in the reported version, and the contrast would be negated.
And if John is elderly and punctilious he might have meant something entirely different when he said “I may go to Paris next week”—viz., permissive may:
I have permission to go Paris next week.
He cannot express this meaning with the ’past’ form might, and the contrast is restored.
As you probably know, the ’past’ form of a verb does not always express past tense: it may also express what CGEL† calls ‘modal remoteness’: some kind or degree of non-reality, such as counterfactuality, or hypotheticality, or tentativity.
If we had some bacon we could have bacon and eggs, if we had some eggs. —which implies that we have neither bacon nor eggs
And this gets very tricky with the ‘modal’ verbs can/could, may/might, must, shall/should, will/would, because these verbs all have (among other meanings) an epistemic meaning involved with certainty and probability—which is the same semantic domain as ‘modal remoteness’.
So what exactly does it mean when you use the ‘past’ form of a modal verb? Is it a tentative assertion of a present definite eventuality? A positive assertion of a hypothetical present eventuality? A positive assertion of a past tentative or hypothetical? A counterfactual? —You pretty much have to figure it out from context: what is this utterance likely to mean?
And it’s made even more complicated by the fact that the ‘meanings’ and the ‘rules’ and even the words themselves are changing. Even native speakers, with a lifetime of practice drawing correct inferences from complicated contexts, yearn for more regularity and predictability. Over the past five or six hundred years we’ve been gradually supplementing and to a significant extent replacing the ‘modal’ verbs with more conventionally inflected periphrastics like be able to and have to and be going to VERB. At the same time we’ve been collapsing the epistemic senses of ‘present’ forms of modals into their ‘past’ forms, and either redefining the ‘present’ forms or discarding them. For instance, epistemic can with present reference is now restricted to negatives and contradictions of prior negatives; in present positives it is replaced with could. Must, originally a ‘past’ form, completely lost its ‘present’ form, mote, centuries ago; the remaining form is by default treated as ‘present’, and use as a ‘past’ is largely confined to subordinate clauses.
The may/might pair seems to be going the same way as can and must. Mair & Leech‡ find that between 1961 and 1992 use of may declined by 32.4% in AmE, while use of might declined by only 4.5%—less than most modals. (There are similar patterns in BrE, but less marked.) A lot of may’s decline is attributable to a shift of permissive may to can even in formal registers. However, my own experience suggests that the epistemic sense of may has also been seriously weakened. When I write in copy directed to the public that such-and-such a financial instrument may have such-and-such results, the compliance orcs inevitably find it necessary to supplement may with “potentially” or some such redundancy. This doesn’t happen with might, which apparently (at least for the time being) still has a strong epistemic sense of possible but not certain.
So—sometimes may and might mean different things, and sometimes they mean the same thing. But even when they mean the same thing, it may not (or might not) be the same same thing that they meant 50 years ago, or fifty years in the future.
† Huddleston & Pullum, The Cambridge Grammar of the English Language, 2002, the closest thing we currently have to an 'authoritative' English grammar.
‡ “Current Changes in English Syntax”, in Aarts & McMahon, eds., The Handbook of English Linguistics, 2006