I'm not a grammarian, let alone a _serious_ one, but I'd like to make a case for both words being synonymous in that context.

Let's look at the following phrase in which a student asks a teacher for permission to go to the bathroom:

> [1]  May I go to the bathroom?  
> [2]  Can I go to the bathroom?

Sentence [1] is pretty self-explanatory.

Sentence [2], though, could be interpreted as follows:

> [2-alt]  If I have your permission to go to the bathroom, then I am able to go.  If I do not have your permission, then I'm not able to go.  Therefore, I am able to go to the bathroom?  Or, rather:  can I go the bathroom?  

Sentence [2-alt] uses the definition of "can" that means to "be able to."  The student is asking if he or she is able to go to the bathroom.  (if he or she does not have permission, then he or she is not able to go.)  The "with your permission" part was _understood_, if you will.

**edit:** Changed [2-alt] to be slightly more logical and slightly less redundant.