There is a difference, and it's a significant difference. Contrary to popular opinion, the two words are not interchangeable. An individual experiences "amusement" when anything makes him smile or laugh. He finds it amusing or humorous.
Example:
The principal found the comments made by the students very amusing.

Unlike "amuse," "bemuse" has nothing to do with amusing people. You are confused by something when you are "bemused" by it. You might be perplexed or baffled by it.

Example:
The students stared at the stranger bemusedly as she entered the classroom.