Hello, Lariel—and welcome to English Forums. Where both mean 'permission', they are synonymous.
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Mister MicawberWhere both mean 'permission', they are synonymous.I agree that the meanings are practically synonymous, but would caution that they are not necessarily interchangeable. If a person approaches a table in a cafe and asks 'May I ...?' (the ... 'sit here' is often implied rather than said), 'Am I allowed to (sit here)?' would be very unnatural.