What do you have now? Of the 2 you listed, the Radeon is the better card. Only trouble with ATI (the company that makes the Radeon chipset) cards is the numbering system is NOT easy to figure out...sometimes, the lower model cards wil outperform the higher number cards. A 9200 is not *necessarily* better than a 9000, for instance. You have to look at the memory timings, etc. It can get quite confusing. As for the NVidia cards, try to avoid anything with "MX" in the model #. These cards are pretty castrated in terms of video pipeline and other stuff from their non mx counterparts.
A video card is only part of the chain, though...a marginal video card upgrade to a PIII system would be a waste of money, imho. Bare bones minimum in my book for gaming these days is an Athlon 1800xp system w/ at least 256mb of ram, and a Radeon 9000, or GeForce 4 non MX card. The newer games coming out now want LOTS of video power.
As you switch to the more powerful cards, you may have a potential power problem....I've had to replace power supplies in several of my computers when moving up the the 9800s and GeForce 4s. You need a certain amount of power on specific rails of the power supply to run higher end processors and graphics cards...usually a 400w or higher model power supply will work, but not always; some better brands have the power you need in a 300w model.
If you stick with the lower end Radeons and Geforce MX cards, you won't have that problem...but you might not be getting much of an upgrade with those, depending on what you're running now. Let us know what you're running now....