Amuse means to entertain. (see aslo: Amuse vs Divert vs Entertain vs Recreate)
- His jokes still amuse us, even though we’ve heard them all.
- It amused him to think that they were probably talking about him at that very moment.
- I'm sure I'll be able to amuse myself for a few hours.
Bemuse means to bewilder, confuse, or stupefy. (see also: Daze vs Stun vs Bemuse vs Stupefy vs Benumb vs Paralyze vs Petrify)
- The change in tactics appears to bemuse our opponents.
- The sheer quantity of detail would bemuse even the most clear-headed author.