Why does discharging a capacitor give a higher power output than using a battery? 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? 
 A: You are considering perfect batteries, that could maintain their potential under any load (current), but real batteries always have some internal resistance, which causes a voltage drop under load.
Since the internal resistance of a capacitor is much lower, it's able to deliver considerably higher currents.
The reason for this difference is that in a battery there is a latency associated with the chemical reaction to transfer the chemical energy into electrical energy [while a] capacitor is storing the electrical energy directly on the plates so discharging rate for capacitors are directly related to the conduction capabilities of the capacitors plates [Science News for Students].
A: Don't confuse energy and power.  The capacitor almost certainly has a far smaller energy capacity.  What it can probably do is deliver its small amount of energy very quickly so, for a brief moment, its power is higher.  It would be similar to a race between a sprinter and a marathon runner.  The sprinter will make a much faster start but tire much more quickly.  The marathon runner will keep going much longer.  
