Build has a 285 watt power supply, so that limits you unless you are willing to upgrade the PSU as well. If you budget $75 for a new one, you are left with $250 or so. The Radeon 7870 is in there, also the 650 ti on the lower end.
also - the difference between the makers of the cards - some have nicer coolers, memory, memory/gpu speeds, or better warranties.
Hmm... for about $350, you can get a GTX 670 or a Radeon 7970. Either of those will max out 2013's games, and will last on high / some medium settings for a long time to come.
The differences between manufacturers come in two ways: The first is the clock speed of the chip. If you want to overclock your graphics card a little, this won't matter, but if you don't, then higher is better... all else being the same. (For example, GDDR 5 vram is a must have over gddr3.)
The other difference is the cooling. If you are overclocking, this is important; you don't want a reference cooler, as they run hot and noisy when overclocked. The cooler is also indicative of the brand quality - look at some cards in your price range, and ask around as to good brands. (For example, Asus and Sapphire are cream of the crop, while Powercolor and Zotac are, well... not.)
ALL that being said, there are two problems you're going to have.
One, your power supply is crapola. Upgrade that to a good quality 550w PSU. I recommend Antec or Seasonic. (Make sure it has at least two PCI-E power connectors; four is better.)
Second, I can't tell, but that might be a slimline case - if it is, that means it can only accommodate half width (low profile) cards, in which case you're going to have a LOT of trouble finding one you can game on. If it is slimline, I'd recommend upgrading to a $40 NZXT Gamma Classic, or something of roughly equal price and build quality.
That linked PSU brand, Ultra, doesnt have a single review on newegg... sounds fishy. I would get a PSU from a quality brand like Corsair, Antec, Silverstone, PC power&cooling. Buying a cheap one will potentially kill all of your other components.
If you have the $ left over from a psu, the 7950 is the one to go with out of those two. Long legs if you have a CF motherboard and can pick up a second card in a year or two.