You should use up all 2 connectors. You won't tell any negative side effects your PC would have until you play a very intensive game lke Crysis, GTA4, CoD series, latest Battlefield series, Mirror's Edge, etc. The reason that you must use 2 connectors is to avoid any crashes caused by insufficient power for the video card. It's not performance but power stability is the thing.

Functionally, the two connectors both connect to a bridge to the power circuit on the board. If you power supply is using heavy-gauge wiring and has lots of power on the 12V rail connected to the PCIe plug, you will be fine.

The twin 6-pin sockets are for situations where your power supply has very light 12V rails (generally on PSUs with only one 6/8-pin plug, the other is often adapted from one or two molex 4-pin plugs), or on cheap PSUs with light gauge wiring where wire resistance can be an issue in getting the amperage across, or (in terrible cases) where the GPU pulls enough to melt the insulation due to excessive resistance generating heat.

Two plugs is for safety. It is the logical way to go. But you can often cheat it if your equipment is up to par. But only using one increases the chances of catastrophic failure.