PhysX is an entire physics API. PhysX itself is the base standard for it, which runs the non-'eyecandy' stuff that you're talking about. Stuff like destruction calculations, vehicles, hit mapping, animations, ragdoll calculations, etc.
PhysX APEX on the other hand is a subsection of PhysX that deals specifically with the eycandy. It handles things like cloth physics calculations, 100,000+ particle effects, volumetric FX, and debris clutter.
Realistically, without APEX stuff turned on, Nvidia and ATI users will have identical experiences. It will all work the same, no matter what.
With APEX capabilities on though, Nvidia users will have much better graphics due to the fancy effects that APEX enables that I explained above. Gameplay will remain the same, but it will look even better.
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I'm not really all that upset by this. I have the capabilities to continue to use my ATI cards while still taking advantage of GPU calculated PhysX. But that still doesn't change the fact that like someone said above, most ATI users won't go near the 'jerry-rigged' way of getting it to work like I have.
ATI recently overtook Nvidia in the discrete GPU marketshare, which indicates that despite the fact that Nvidia relies so heavily on PhysX and CUDA as sellingpoints for it's cards, consumers are definitely more interested in $

erformance and efficiency than over advertised anti-consumer 'features' that alienate over half of a PC games potential user base.
Make no mistake; PhysX will drastically improve the Arma engine, even without the fancy effects. But it's still going to piss a lot of people off anyways, when there are comparable inter-compatable solutions out or coming out in the near future that could have been used instead.