I wouldn't say it is appreciably better for every game, not unless you are playing at an extreme high resolution or across multiple monitors, or if you are playing with stereoscopic 3D. If you are running 1080p or lower, the 6870 will deliver a solid 60 FPS in most titles on the ultra high settings. Only the extremely GPU demanding titles (Crysis, Metro 2033, BF3, Witcher 2) will see any performance boost, and even then it isn't that large, usually only 3 to 10 FPS more. The CoD games are not GPU demanding titles, either card would do fine unless he's playing at 2560x1600. Go with whichever one you can afford, the 6950 would be slightly more futureproof I guess, though perhaps not necessarily worth it if you have to spend a huge amount extra for it. As said above, don't bother with the 2GB version unless you have, or plan to upgrade to a display higher than 1080p, or intend to game across mulitple monitors.
 

Ninja Pants

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I assume your question is concerning the 1GB 6950 as this seems the viable comparison. Here is a direct comparison: http://www.anandtech.com/bench/Product/290?vs=331.
For resolutions below 1920x1080 you are unlikely to see a big improvement with 2GB VRAM vs 1GB VRAM as your never going to be bottlenecked by the display buffer width. At those resolutions best bang for your buck would be CF6870's which come in around the same price as the 2GB 6950 but will outperform it on a single monitor system.