First, the meaning is different.
1, possible: (possible but uncertain) possible, possible.
Maybe: (May's past tense is used as an indirect speech) Maybe, yes.
Second, the usage is different.
1, May: May can indicate possibility, which means "maybe, maybe". It can be followed by completion or infinitive, which means "maybe it has it, maybe it is [will]" and so on. May is used in imperative sentences to express wishes by adding verb prototypes.
2. May: May can be used as an auxiliary verb, collocated with a notional verb, to express small possibility, permission, wish, hope and request, and can also be used in clauses expressing hope or purpose. May is also the past tense of May, which is used instead of May in questions to show politeness.
Third, the emphasis is different.
1, may: it means asking for permission, and may is a little blunt.
2. Maybe: Maybe it means asking for permission, maybe the tone is more euphemistic.