Why use capacitors to store energy?
Any actual source is going to diverge from the behavior of the ideal sources that we use in circuits. The first rough estimation of a more realistic source is to assume that there is an internal resistance. This would be modeled as having the source have a resistor in series (for a voltage source) or in parallel (for a current source I think) between the source and the output terminals. In this way, we see that power gets siphoned off from the source and expended into the internal resistance. It is thus easy to see that this will restrict high current draws on a voltage source for example. That is, if we place a low resistive load across the voltage source terminals, the effective resistance will be dominated by the internal resistance. So even as we place a smaller and smaller resistive load, the drawn current will change very little due to the comparatively larger internal resistance.
Another effect of the internal resistance is that it behaves as a dampener in a LC circuit. If I place a function generator that has an output impedance of 50 ohms onto a LC circuit, its behavior will be like an RLC circuit, where the R element comes from the output impedance/internal resistance of the source. This affects the supplied voltage/current relationships of the LC circuit. For example, it would shift the resonant time constant of the circuit from what it would have been with an ideal source. So the internal resistance can also affect the transient behavior of a circuit.
This is one reason why fast transient and high current draws can be better taken off of capacitors since they can be precharged with enough charge to drive the necessary fast current spikes. Capacitors also are simple filter elements. Not only can they provide current compensation when the source lacks the current for a sudden transient, they also can act as a simple low pass or high pass filter at the same time. In a simple power supply circuit that uses a transformer to rectifier to filter to voltage regulator, the filter stage is just a resistor and capacitor. Both elements work double duty as the capacitor and resistor first act as low pass filters to help smooth out the rectified AC wave from the rectification stage. In addition, the capacitor also helps by being a charge resevoir and the resistor controls the bias current that biases the zener diode that acts as the voltage regulator.