The two words are closely related and can be considered synonyms of each other. Most of the time, the two words are interchangable. There are some situations where you would use one over another. Amuse usually carries a light or playful meaning with it--just like how you would find a joke "amusing" (although using entertaining in this case would still make sense, amusing would usually be used). Entertain, in some instances, can carry more of a hosting meaning with it. Perhaps you start a game to entertain guests that are over at your house. Another use for the word entertain is when a person chooses go with an idea. You can say, "I will entertain that idea for now."

entertainment provides amusement