Sometimes they mean the same thing. For example, these are the same: "I might go shopping today" "I may go shopping today" But "may" has another meaning as well: "You may go shopping today" - this means "You are allowed to go shopping today". Sometimes it is the tone of voice that makes the difference. "may" can have different meanings depending on how you say it. If you say it with authority it means "allowed to". If you say it more like a question, it means "might". Also: "You might not go shopping today" - this is a suggestion "You may not go shopping today" - this is a command, you are not allowed to go.