Batteries generally have fairly short lives; this is certainly true with lithium batteries, whose high performance characteristics are offset by the fact that they're really only just barely stable from a chemistry standpoint. There have been a number of spectacular fires resulting from damaged or overcharged lithium batteries; as they age they become less stable; most lithium battery packs include little special-purpose "smart power" chips to keep them from exploding. These chips monitor their temperature, charge state, and other parameters to keep them stable.
For that reason, I imagine that supercaps might find uses in satellites and other applications where maintenance is impossible.
Another application for supercaps would be in hybrid cars, where you have a regular engine supplying power, and an electrical power source supplying additional power for acceleration, and storing power from regenerative braking.