Let's say you charge a capacitor with a 10V battery. The capacitor will have a potential difference (pd) of 10V once it is fully charged. If you then connect a light bulb to the capacitor, the pd across the light bulb will also be 10V until it decreases exponentially to 0 as the capacitor discharges.
How is this any different to connecting a variable resistor in series with a 10V battery and an identical bulb, and then increasing the resistance of the resistor so that the pd across the bulb decreases at the same rate as it did with the capacitor?
I've read that the capacitor produces a greater power output, but how can this be true? Using a capacitor rather than a battery wouldn't cause the pd across the bulb to be greater, nor would it cause the current through the bulb to be greater (the current with a capacitor would be the same initially as it would be were the bulb connected to a battery, but it would decay exponentially along with the pd).
Therefore, how does discharging a capacitor result in a greater energy and power output than using a battery?