"Must" is much stronger than "may".
You can use "must" to indicate an obligation: You must pass your driving test before you can drive a car on public roads.
You can use "may" to indicate permission: You may smoke in here.
You can also use "must" when you make a firm conclusion based on evidence: Jack says he was in library at the time, but Mabel says she saw him in the dining room; one of them must be lying.
"May" is then used to indicate that you're not sure if your conclusion is right, but it's possible: Jack may have murdered Lord Arlingworthy; he had the opportunity, but on the other hand he had no motive.