Mousemonkey :
So why do AMD user get all up in arms about PhysX then?
Probably because PhysX is a relatively high level API that actually could be adapted to individual vendors' GPU architectures if Nvidia wanted to license it to its competitors for a price they could all live with. One of the worst pages in PhysX's history is the time where Nvidia disabled GPU-based PhysX if non-Nvidia GPUs were detected in the system. Imagine how happy people with a HD6970 were when the GT4xx/5xx they added for PhysX refused to work due to PhysX detecting AMD GPUs in the system.
Another thing is that PhysX interest is not uniform even across Nvidia's own fan base since not everyone wants to sacrifice FPS for physics-based eye-candy and clutter.
On the other hand, Mantle is something AMD can do mainly because they happened to nail all major next-gen console graphics contracts using the same GPU architecture and that puts them in a unique position to offer the same low-level API across multiple high-profile platforms. Console developers are used to having lower-level access to hardware to push the boundaries of what is possible on a given platform so AMD would have had to provide ways to do it anyway. Might as well capitalize on that.
The benefits of PhysX are a lot more disputable than the benefits of Mantle and Mantle probably won't have much of a life outside games that get backported from consoles since PC game developers likely won't feel like optimizing both a DX or OGL back-end and a Mantle one specifically for GCN-based GPUs. Forcing Mantle-based console backports to use DX/OGL when Mantle is available might end up feeling like driving a race car with the handbrake on.