Hi,
I think the only difference is that "may" is less common in everyday conversations, in informal contexts. It belongs to a (slightly?) higher register.
The fact that "may" implies that something is more probable is a well-known urban legend in the ESL field, which Michael Swan helped to spread. Maybe he was even the one who started it...
If you go ask some native speakers, "What's the difference between these two?"...
- The bridge might soon fall to pieces
- The bridge may soon fall to pieces
...I don't know how many people will tell you there's a difference in terms of probability. Maybe no one. Maybe most of them will just say "No idea. Is there a difference?"