80 PLUS Bronze is an efficiency rating. You can have a POS 80 PLUS power supply that has a noisy 12V rail or won't run multiple cards, or even one card in some cases.

1x is even fine. Mining doesn't need much bandwidth to the card. (You can even use the physically smaller slots with adapters or modifications.)

The only considerations are physical compatibility, GPU-chipset compatibility (5000 and up cards won't run on nv chipsets), sufficient cooling, and a good power supply. If you are only going to run one GPU, you will probably be fine with either computer and there wouldn't be an advantage in either case, unless you are worried about power consumption.