Although both words once held similar meanings—to beguile, delude, muddle, stupefy—amuse, by the early 1600s, came to mean to divert attention with entertainment and, by the 1700s, to entertain, to tickle.
Bemuse, however, has retained its original sense and still means to puzzle, confuse, bewilder. Though the two words appear similar and stem from similar etymological roots, and while their differences may bemuse writers, do not mistake the two words as synonymous. They are now unrelated.