Principle difference is in the yeast as indicated above, but also the hops. American Wheats seem to be a pretty wide style. I tend to think of them as either a Witbier or a Hefeweizen, without the distinctive yeast character (ie a cleaner American yeast). Sometimes they're spiced, sometimes they're not. Sometimes they're low on hops (while Hefes are always low on the hops), but sometimes they're fairly well hopped, somewhat like an American Pale Ale (or perhaps a Blonde Ale), but with a bunch of wheat in the grainbill.
So yeah. I'd say key difference is the yeast- you can't make a Hefe with clean American yeast and you can't make an American Wheat with Hefe yeast. But where Hefeweizen is a pretty narrow and restrictive style, American Wheats have much more room for interpretation.