I've been wondering this myself. The differences seem superficial or semantic to me. Unfortunately, there are few works out there that bother to address non-monotheistic theology. The ones I have managed to track down and read have not been particularly enlightening in drawing a difference between the two.
Thinking about this from the standpoint of the history of philosophy, though, it was once a "given" in philosophical thought that all things had a sort of inner spirit or purpose for being. When certain "advances" in human civilization occurred (e.g. the Enlightenment), the ensouled view of matter was thrown out in favor of a disenchanted and mechanistic one. It was no longer proper to speak of all things as having a teleological purpose. So in at least some contexts, animism does not particularly describe a religious belief at all, but a philosophical position. It seems to me, though, that when animism is spoken of in a religious context it's pretty much identical to polytheism. But I don't know.