They are nearly the same. "The ball was a source of entertainment for the children." "The ball was a source of amusement for the children." I can't think of any better sentences with those forms of the words, but the verb forms work well. "The television show was amusing." (funny) "The television show was entertaining." (might be funny, but not necessarily)

I think they basically have the same meaning these days. If you want to look at the origins of the words in french, amusement sort of comes from the idea of being distracted or deluded. http://www.etymonline.com/index.php?term=amuse Entertainment had more the sense of a public performance http://www.etymonline.com/index.php?term=entertainment&allowed_in_frame=0 Again, by now they are pretty much interchangeable.

As far as I know, entertainment is any activity you can do to amuse yourself. Amusement is what you get out of being entertained. So amusement is the product of entertainment :) Anyone correct me if I'm wrong!

I would like to answer your question through another question. May I ask you the difference between spiritual pleasure and artificial pleasure? If you know means you know all about amusement and entertainment. Actually amusement is our spiritual pleasure which is autonomous. Entertainment may not be a pleasure. It may also be a serious consideration. But generally, entertainment is taken as a made pleasure. Hope it will be helpful.