The 7950 X2 is about 24 GFlop/s processing power (http://en.wikipedia.org/wiki/Comparison_of_Nvidia_graphics_processing_units#GeForce_7_.287xxx.29_series
), the HD 6770 is 1360 GFlop/s (http://en.wikipedia.org/wiki/Comparison_of_AMD_graphics_processing_units#Northern_Islands_.28HD_6xxx.29_series
) so the ATI card is about 500x the theoretical power of your 7950 x2. Because of the difference in architecture though that's not a precise comparison but still, it's considerably better.
The ATI card should do audio out over HDMI too and since it's a single card rather than dual PCB it should be quieter, cooler and more efficient so if you really don't want to stretch your budget $45 to one of these (http://www.newegg.com/Product/Product.aspx?Item=N82E16814121641
) then use that initially. ATI cards don't play as well with linux so you could be looking at issues there if you ever went down that route.
But, neither is an ideal solution for a purely media PC but try them out for yourself and see I guess.
A passive card is really what you want though and the Geforce cards from the 200 series can do lossless audio out over HDMI which again, is something you should want. I had a passive cooled GT240 and it was perfect for a media PC.