This may not be a easy question but I am torn and not sure which one I should buy. It will be used for some gaming not much, mostly surfing the web, downloading & watching videos/movies.
The PC it will go into is a Dell with these specs
AMD, Athlon Duel core, 2.4ghz.
3GB Kingston DDR2 800, Ram
500W PSU (which can be upgraded if need be)
Windows Vista, 32bit.
On board sound.
Sata 3.5, 7200RPM, 16mb 250GB hard drive.
Both cards are great, but keep in mind your am2 x2 will be a bottleneck for both of them. I've got a heavily overclocked X2 240 with DDR3 OC'd to 3.35GHz with a 4870, and even that can't push the 4870 anywhere near as high as it was with a Q9650 at 3.6GHz (it's a full 4,000 points lower in 3dMark06, for example). That's a newer Athlon II (I suspect you have an Athlon X2) and almost 1GHz faster than what you have so you can see you'll have an issue with the processor feeding both cards.
That said, you'll have something to use in an upgrade later when, say, you upgrade that processor to a quad (if you can). I'm pretty fond of the 5770 for a mid-range / budget card as I've been able to push them to higher bench scores than the 4870 and you get DX11 and EyeFinity (I'm VERY fond of EyeFinity). And, the 5770 actually has enough power to use three 1680x1050 or 1440x900 monitors in most games.