So why aren't supercapacitors used more often?
Low energy density: for a given amount of energy, they take up more space than almost all battery technologies
High leakage: this document has some nice curves. It looks like they've lost half the voltage after a week.
Bad discharge curves: normal batteries might give you 80% of the power before they've dropped 20% voltage. Capacitors give out power evenly across their exponential curve. So either you end up using a small amount of the capacity before the voltage drops too low for your application, or you need to mess around with boost converters on the output to keep it matched.


Edit: Peter Smith's link highlights some good applications. They're not widely used in cars yet because of the density and leakage issues, but for industrial equipment they're a good fit.