For the most part, the GeForce FX 5600 256 MB will have a benefit in future games, where heavy texture usage will be common (e.g. Doom III), but for now I lean towards the Radeon 9600 Pro. Why? Higher image quality - and the performance doesn't take much of a dive as you enable anti-aliasing, anisotropic filtering, or both. Plus speaking of anisotropic filtering, ATI's implementation yields higher image quality, due to the capability of a higher number of texture samples used (16 vs. NVIDIA's 8).
Furthermore, it looks like you're talking about the non-Ultra 5600. The 9600 Pro will easily win over that card.
Speaking of compatibility, my friend owns an A7N8X Deluxe and has had no problems with his 9500 Pro.