They have the same hardware units with regard to pixel pipelines, vertex shaders, etc... Get the one with the higher core clock and memory clock. Core clock is more important then memory clock however, and more important then memory amount in the case of graphics cards of lower caliber.
Having said all that, I'd have to think they'll be quite similar in performance, but the 9600 XT will probably win out because of higher clockspeeds - on the memory especially.
But like starcraftfanatic said, you can buy a new card that will absolutely destroy them both.
Of course, you'll need an AGP card if you have a machine that'll take an X1950 XT. But for under $100 you could get a used 6600 GT on ebay that would slaughter both of the options you're putting forward.
Having run both those cards, the 9600 Xt will slaughter the X1050 in raw performance. Although the X1050 has more features.
6600GT might do well, but I had more luck with the 9600XT. Good competition between my brother PC and mine, back when they where new.
He had the following:
Athlon XP 2500+
Soyo Dragon Black Mobo
Athlon XP 2400+
abit nForce 2 Mobo
968 MB ram
9600Xt (non third party)
In that case I ran just slightly ahead of him. I thought it was kind of funny though, he though he was going to get me in performance. I think some of it may have had to do with the motherboard.