May is used when one asks for or gives permission.
Examples:
Can is used when one expresses or shows ability to do something.
Examples:
Read and study the dialogue.
Joan went to the library to borrow a book for her Reading class. (Joan is knocking at the door)
Mrs. Jose: The door is open. You may come in.
Joan: Good afternoon, Mrs. Jose.
Mrs. Jose: Good afternoon, Joan. What can I do for you?
Joan: May I borrow a book which has a fable written by Aesop? May I take the book home? I can copy the fable tonight and share it with the class tomorrow.
Mrs. Jose: Here, you may take this home.
Joan: Thank you, Mrs. Jose. May I ask help from you again next time?
Mrs. Jose: Yes, of course you may.
Joan used the expressions may I borrow and may I take for permission. She used can copy to show her ability to do something.
Can and may are followed by the base or simple form of the verb.