Strictly-speaking, "may" is supposed to be for permission, but "can" has completely taken over, unfortunately. "May" also indicates something may possibly happen, has the possibility of happening. "Can" can mean something could possibly happen, but it doesn't mean it's "likely" to happen; "may" is more for that.

I would just add that "may" is rarely used now in regards to "permission". Very rarely will you hear, "may I have another slice of pizza?" Come on now - it's pizza! Don't say anything - just take it!!! The word "may" is now mostly used to denote some probability as in "it may rain tomorrow".

Yes, and you nailed it on the head when you spoke of permission vs. capability. Here's a conversation that I hear take place on a daily basis at public school that illustrates this point while also demonstrating that this distinction is a formality. Student: Can I go to the bathroom? Teacher: I don't know, can you go to the bathroom? Student (exasperated): May I go to the bathroom? Teacher: Yes but first tell me: how many years have you been asking me that same question and getting the same answer? Student: Three years. The teacher fully understands what the student means but feigns confusion for fun as much for its teaching value. The student has been through this many times before but still makes the same mistake because this situation is probably the only situation where the student is being asked to make a distiction between the two words. Part of the humor in this situation comes from how the phrase "go to the bathroom" is meant as "walk to the bathroom" or "leave the class to visit the bathroom" but when the question is rephrased as "CAN you go to the bathroom?" it sounds like a question a doctor or nurse would ask you. The student's peers in this situation interpret the teacher's playful rephrasing as an informal version of "Are you capable of defecation?" or "Are you capable of urination?" I think this double-entendre is the reason that this specific situation is so often singled out as a chance to teach the difference between "can" and "may". Many other situations occur where people use "can" when they really mean "may." For example: Can I have it Mommy? Can I help you? (this one could go either way actually) Can I take your coat? Can I go to the Movies tonight? Can anybody else offer some good examples where "can" replaces "may." With this last use of "can" I really am using it in the sense of "are you capable" despite the fact that I am sure other English speakers are "capable" of providing more examples. I also realize I have no authority to tell them what they may and may not do. What I'm really saying is "Will someone else offer some good examples where "can" replaces "may."

yes the difference is may=probably i doing some thing . can = the Ability to doing something