May - asking for the permission
Can - stating or asking of the ability
The distinction is observed mostly by English teachers. Nevertheless, "may" signifies permission, and "can" signifies ability or possibility.
Thus, if a street intersection had a sign saying "No left turn," and someone asked "Can I turn left here," a very pedantic person might say "You can turn left here, but you may not, because it's against the law."
Or, if someone said "May I get a pumpkin pie at the bakery?" a possible reply would be "You may if they have them, but I don't think you can, because I think they only have them at Thanksgiving."
Roughly speaking, "may" is the choice to do do something, and "can" is the ability to do something.
may = poder (con permiso, decisión)
can = poder (con abilidad)