Centering:
As CampyBob pointed out, a single pivot brake needs tools to center (rather than a firm nudge with your hand). This is because there is a two sided spring (left-right) held by the center bolt. (In contrast, the dual pivot brakes have one spring for one side, another for another, and they are mounted on a piece which you can tilt, tool-free, from one side to the other). If the center bolt on a single pivot brake is tilted to one side (say the right side of the spring is tilted down towards the rim), the corresponding brake pad will rub. You can use your fingers to twist the brake back in place till you use the brake again, but as long as the center bolt is tilted, the brakes will always recenter around that "tilt".
Of course, if you have the brake centered properly, and the center bolt is tightened sufficiently, it will stay centered.
Advantages Single Pivot:
1. lighter (less stuff in brake)
2. simple (less stuff)
3. cheaper (less stuff)
Disadvantages Single Pivot:
1. can't re-center brake during a ride
2. can't re-center brake without tools if you put in a wheel that is slightly "off center" to the brake's current setting
Braking Power:
There is actually a difference in braking power, simply because the dual-pivot brakes have more leverage. The same difference exists between "normal" cantilever brakes and the "V-brake" setup. The normal Cantilevers have rapid pad movement but less leverage. The V brakes have tons of leverage (they were so strong compared to normal brakes that many companies had to redesign their frames) but the pad takes a while to move. In behavior, normal cantis=single pivot and Vbrake=dual pivot.
It is not as apparent in current sidepull brake designs since all brakes seem to work pretty well. The extreme "low braking power" example for single pivot brakes is the old Aero-Gran Compe. The brakes had such little leverage that slowing firmly was an adventure in itself. The rider got instant pad movement but could not exert a lot of force on the pads. It didn't help that the pads were small, of poor material, and of unique shape with no real aftermarket pads available.
For ultimate braking power, both types of brakes work fine - they'll all lock up a wheel with high/normal hand pressure (except the aforementioned Aero Gran Compes). Of course it isn't really an issue since using max braking power simply locks up a wheel and skids a tire or dumps you on your face.
So the real question is which allows you to use the maximum effective brake pressure? In that respect the easily-modulable dual pivot brake wins. Hence Campy puts the heavier but easier to control brake in front, and the lighter but a little less controllable brake in the rear. Locking the rear is not as dangerous as locking the front.
hope this helps
cdr