My non-definitive answer is that "plush" is a term commonly used in U.S., but is not so common elsewhere. I believe they refer to the same thing. Technically plush suggests softness, and I suppose it's possible to have a stuffed toy with a hard exterior.

Stuffed animals are what taxidermist make (made from real animal skins, stuffed and mounted to produce an ornament or trophy); plush is a kind of fabric with a fluffy surface; at least that's how I would understand the terms, but in America it seems they are used for what in Britain would be called "cuddly toys", "soft toys" or just "teddies".

They're pretty much the same. I think "plush," as Timo said, refers to the material, so a "plush toy" is any toy that is soft, but not necessarily an animal. "stuffed animal" is more of a description of what it represents and is inside, rather than describing what the external is. In practice, they are the same item in 99% of the cases, but in theory they could be different. I think that's where the difference lies. :)