the main benifit for super-capacitors is they hold a significant amount of charge, while taking effectivly no time to reach full capacity, say you had a system of sensors you wanted to power for about half an hour, but only wanted to apply power during a single button press,
only way you could get the capacity in that time frame without some time controlled switch would be with a supercapacitor,
this is just an example, and probably not the best one either,
small details: high capacity, but self discharge is in days at best,