They are exactly the same, for the most part, except Nvidia cards can do PhysX, while AMD cards can't.
The other difference will be in the drivers and the settings/technology offered. In driver settings, there are some different antialiasing settings for each. Only Nvidia cards can force Ambient Occlusion. Only Nvidia cards offer Adaptive VSync.
In the latest generation cards, AMD offers Graphics Core Neutral, which entirely turns off the GPU when not in use (when the monitor is off). Nvidia's latest cards offer Turbo Boost, which self-overclocks the GPU by dynamically increasing clock speeds and voltages when a game allows it.
There's a few other things here and there, but that's pretty much it.