Why is my new iPad 3 only showing 768 x 1024 screen resolution?

It is likely that the app you're using doesn't support Retina display resolution. App developers have to support the retina display resolution through update releases. Apps don't automatically run on 2200+x1500+ screen resolution. They only do so if the developer supports it.

The likelyhood of getting a Retina update for the app would depend on how active the developer is.

Many sites/apps are still not optimized for the iPad 3 Retina Display.