In the "may" case, the possibility is about the state of the world: something may happen because the state of the world allows for it. My father and I both flew to a family reunion last year, but didn't coordinate travel plans; upon learning that we were going to share a flight, either of us could have said "we may be sitting next to each other". That is: without having discussed our seat assignments, it was epistemically possible that we would have adjacent seats.
In the "can" case, the possibility is about the ability to do something (which, of course, necessarily includes the state of the world, but there's more to it than that). Consider that same trip, but in a counterfactual world in which Dad and I were specifically trying to sit next to each other. If we were on the phone while picking our seats, he have said "row 13 is empty; if I pick 13a and you can take 13b, we can sit next to each other". That is: it is theoretically possible for us to sit next to each other on the flight.
In the "may" case, where we had already selected our seats but hadn't discussed what those selections were, "can" would be inappropriate as we didn't know whether it was still theoretically possible to arrange that (eg., maybe the flight was already fully booked, so neither of us could change our seat to sit next to the other).
"We may see you tomorrow" carries that same "may" possibility: we know (or have reason to believe) that we will be in the same general area, so it's possible that we'll see each other. For completeness' sake, that sentence can also mean that we would like to see you, but it may not be possible (eg., "we have a ton of housework but, if we get through it, we may see you tomorrow" carries much the same epistemic possibility sense, but it also implies that we want to see you tomorrow).
"We can see you tomorrow" carries the "can" possibility from the flight: it means that our schedule is open (or flexible enough) that there is room to see you - it is theoretically possible to arrange the world such that we can expect to see you tomorrow.
... which is to say that the other answers are right: in those sentences, "may" means that it's possible and "can" means that the event is highly likely (probably because both parties are working towards that goal).
So, on to your repeated comment:
There is, however, a rather formal use of may where the meaning of 'possibility' is the same as for can. Thus in Transitive verbs in English may be either active or passive, can could be substitute for may with no change of meaning. This use of may is typically found in formal contexts such as in academic writing. This difference between the 'possibility' senses of can and may is discussed in section 121.
This is a whole different scenario to the "factual/theoretical" possibility dichotomy. Academic writing is a specific, formal context; in such contexts, it is common for words to take highly-specific meanings which are related to - but not quite the same as - the words' more common meanings.
In common, vernacular English, "may" talks about possibility while often carrying a "permission" shading where "can" carries a "physically capable" one (consider the old saw about the student who asks the teacher "can I go to the bathroom?" to which the teacher responds "I don't know; can you?"). Further, "can" carries shading of intentionality ("we can see you tomorrow" carries a feeling that we want to see you tomorrow, even if it's just to get some unpleasant task taken care of). Neither of those shadings are appropriate in most academic settings - a photon needs no permission to scatter through a slit, nor does it do so with any amount of intentionality; it simply does so (except when it doesn't).
Thus, formal academic writing (as I understand it) bypasses the difference, choosing to use "may" for all non-deterministic, non-essential events ("the photon may do X" or "the subject may sit down", but "the subject must lie down in the MRI"). In a strict "theoretical possibility" sense, then, "can" can replace "may" in formal academic writing.
Compare RFC 2119, which sets out formal definitions for "may" (and a few other terms) in regards to a formal context (specifically, for other RFCs - Requests for Comment, the way that most of the standards that underlie the Internet are written). Of particular note, "can" is not defined, since its definition would necessarily either overlap with "may" or be too fine a gradation (and, thus, cause argument over which should be used in a given situation).