There are really two separate parts to your question:
what is the difference between a supercapacitor and a battery?
why does graphene make such good supercapacitors?
A battery and a supercapacitor work in very different ways. A battery generates electricity using a chemical reaction. As the reaction procedes a current is generated, and once the reagents have all been used up the reaction stops and the battery is exhausted. The current that a battery can generate depends on the rate of the reaction and the internal resistance of the battery and is generally limited. However it's easy to store a lot of energy as chemical energy, so batteries can produce a lot of power before they run down.
By contrast a capacitor stores electrons directly. The sort of capacitor in the computer I'm using to type this is basically a parallel plate capacitor. Capacitors like this have virtually zero internal resistance, so the charge can flow off them at very high speeds (i.e. high currents) but each capacitor can store only a small amount of charge, so they store much less energy than a battery.
A supercapacitor stores electrons directly, like a parallel plate capacitor, but using a different mechanism that can store much more energy. Carbon based superconductors use a porous carbon matrix with a liquid electrolyte, and they store the electrons in an electric double layer at the carbon surface. The amount of energy this type of supercapacitor can store is comparable to a battery, so they combine a high energy density with very low internal resistance.
Exactly how graphene improves on existing supercapacitors I'm not sure. I imagine the details will be kept secret until all the patents have been acquired! From the articles I've seen on the web I'd guess it's a combination of the very high surface area that can be achieved using graphene electrodes. Obviously the greater the carbon/electrolyte contact area the greater the energy that can be stored in the double layer. In addition graphene is a far better conductor than the amorphous carbon used in existing supercapacitors.
If anyone is still interested, there has just been a paper on the Arxiv describing graphene supercapacitors. See Graphene based Supercapacitors with Improved Specific Capacitance and Fast Charging Time at High Current Density and the summary at the Arxiv blog.