To me, "permitted" sounds more formal. It also sounds like a temporary thing as well.
"Betsy is not permitted to go in here." sounds like Betsy needs the "permission" of someone in order to do so.
"not allowed" sounds like a permanent thing. Kids aren't "allowed" to to drive a car. The man wasn't permitted to go backstage to meet his favorite musicians.
All in all, they're pretty equal! I hope I explained the technical differences as well as a native speaker can; in conversation, either will work smoothly :)