Formally, might is the past tense of may. In situations where the past tense is required, only might may be used:
Correct:
He said he might go.
Subordinate clauses in English must be in the past tense if the main clause is in the past tense, so this is correct.
Incorrect:
He said he may go.
Here you cannot use the present tense may with the past tense main verb said.
However, in the present tense may and might can be used interchangeably, meaning that all of the following are correct:
He says he may go.
He says he might go.
He may go.
He might go.
Note that could is the past form of can, and might is the past form of may.
Past forms of these words are used in subjunctive and conditional constructions.
I can go to the cinema is a statement that you are able to go without any external conditions being in the way. (But the statement stops short of making a commitment: namely that you will go to the cinema.)
I could go the cinema. has multiple interpretations. One is that it's an incomplete conditional thought. You could go to the cinema, if what? It can also be uttered by someone who is in the middle of making a decision. What should I do tonight? Hmm, I could go to the cinema.
Quite possibly, in this kind of reasoning, the speaker, to some extent, externalizes the internal conditions on which the decision hinges. It is not simply true that the speaker can go to the cinema, because that is only possible if he doesn't choose some other mutually exclusive activity for the evening which precludes going to the cinema. That may be semantic the basis for why the conditional-making past participle is used for such statements.
I could go to the cinema tonight or I could go clubbing. I know! If I catch an early movie I can go to the cinema and I can go clubbing.
Now about may. I may go to the cinema is very similar to I can go the cinema, but as a native speaker, you know the difference between can and may being that between ability and permission or possibility.
Furthermore, modern English, the semantics of can stretches to cover that of may (but only in the area of permission, rather than possibility). Children frequently ask grownups permission using can I rather than may I.
I may go to the cinema has at least two possible meanings. One is that the speaker's privilege for that outing depends on permission from some authority. I can go to the cinema can still imply that, depending on the context. For instance, it obviously does in My dad said I can go to the cinema tonight.
But I may go to the cinema also has another meaning: that of possibility, and it means that going to the cinema is on the speaker's short list of possible activities. If an adult states that he or she may go to the cinema, of course we assume this interpretation, and not that the adult has permission from someone else. And I might go to the cinema means approximately the same thing.
The difference between I could go the cinema and I might/may go to the cinema is that the former is associated with reasoning about conditions or alternatives, whereas the latter is just a statement of possibility. The former statement informs us about a decision-making process going on inside the speaker, whereas the latter statement informs us that it is possible that the speaker will later be found at the cinema.
Something might happen and something may happen are not exactly the same, because might is used when conditions are attached. For example, if you lean over the rail, you might fall is more correct than if you lean over the rail, you may fall because you may fall states a possibility which is not conditional on anything. The verb might can substitute for may in expressing a pure unconditional possibility, but the reverse isn't true.
Best Answer
They both indicate that something is possible, but something that may happen is more likely than something that might happen.
So you may go to a party if Matt Damon invites you, but you might go to a party if your least favorite cousin invites you.
As stated by @ Anonym, might is the past tense of may. So you have to use might when you are referring to the past.
For example, even if it's likely that you went to a party last night, I shouldn't say, “you may have gone to the party’; I would say, “you might have gone to the party.”