As verbs the difference between bemuse and puzzle is that bemuse is to confuse or bewilder while puzzle is (label) to perplex (someone).
As a noun puzzle is anything that is difficult to understand or make sense of.
After listening to the confusing lecture, the college girl was bemused.
When reading, the librarian usually wore a bemused expression on her face.
During the festival, I was a little bemused by all the noise at the park.
Footprints in the snow puzzled the hunters who mistakenly assumed they were alone in the forest.
Puzzled by the equation, the student raised her hand and asked her teacher for assistance.
Without a map to guide their trip, Lewis and Clarke were puzzled when they reached a fork in the path.