My family owns both of these desktops. I will be able to pick one to use primarily, and also put in a video card I own. (It's old- Nvidia 8600GTS, but probably better than the standard non-dedicated Intel graphics, right?)

My question is- which one should I choose? I want to play Sims 2 with lots of custom content. Maybe Sims 3 as well.

Desktop #1 (This one has some sort of temporary overclocking functionality built in..)

Processor: IntelŪ PentiumŪ Dual-Core G620 (2.6GHz, 3MB Smart Cache)
Hard Drive: 1TB, 7200RPM Serial ATA 3.5" Hard Drive
Total Memory: 6GB (2GB & 4GB) PC3-10600 DDR3 1333MHz SDRAM UDIMM Memory
Graphics: IntelŪ Graphics Media Accelerator HD

Desktop #2

Processor: Intel Core i3 (i3-540) 3.06 GHz
Hard Drive: 500GB, 7200RPM
Total Memory: 4GB (2GB & 2GB) DDR3
Graphics: Integrated Intel HD Graphics

Any chance you can yank the RAM out of #1 and put it in #2? #2's CPU is significantly better than #1, and the hard drive doesn't make much difference for sims (though you might prefer to yank out the 1TB hard drive too for your own purposes).
The 8600 is indeed much better than the Intel graphics, and should be able to play TS2 well as long as it's in good condition. I wouldn't suggest trying to play Sims 3 on an 8600 GTS - it's definitely not supported, and will probably crash a ridiculous amount. Save up for a new graphics card before buying Sims 3.