Capacitors store energy in an electric field, resulting in a linear dependence of stored energy to cell voltage. You can also see this from the capacitor equation for stored energy Emax [Ws] = 0.5 * C [F or As/V] * U^2 [V^2]

Capacitors also do not age by cycling but by holding them at high voltages at high temperatures.

A should can see, capacitors are rated in Farad or Ampere-seconds per volt: it means that a 1 F capacitor will take 1 second at 1 A to charge to 1 V.

Batteries on the other hand depend on a chemical reaction which happens at one specific voltage - as long as there is enough chemicals still available to react, the voltage barely changes.

Batteries are rated in Ampere-hours, as their voltage is stable, meaning that at their given voltage, you can draw 1 Ampere for 1 hour from a battery rated at 1 Ah.

Batteries age by cycling, as the reactions are not 100% reversible.