The extra four pins on the ATX 2.0 connector provide more power to the card via the PCIe connector than the old standard ATX and AGP connectors provided, but sometimes this alone isn't enough to power the card.
For the 6600GT, the extra power provided through the PCIe interface is enough power to run the card (the notice that PCIe version doesn't need an extra power connector on the card, while the AGP version does).
A card like the PCIe 6800 Ultra, on the other hand, still needs the the power provided by the 6-pin video card connector from the PSU because the extra power provided by the 24-pin connector isn't enough to power the card completely. Still, it is a better solution than the two molex connectors that the AGP version needs for power.