What's the difference? My understanding (perhaps an incomplete one) was that they worked essentially the same way, just optimized for different things (power density vs energy density).
Capacitors and batteries work very differently. Capacitors are made from two conductors separated by a non-conducting dielectric and rely on the behavior of electric fields to charge and discharge. Batteries use the energy from chemical reactions to produce electricity, and some batteries allow the reaction to run in reverse in order to store power.
Wikipedia says that some subtypes of supercapacitors use chemical reactions, too.