A true capacitor stores its energy in an electric field (akin to a magnetic field for a magnet). A battery cell stores its energy by creating a chemical that would happily decompose into a more stable chemical.
If your question, though, is “What is the difference between an electrolytic capacitor and a rechargeable battery cell?” then the above difference is no longer the case.
Both of them might consist of two electrodes dipped into an electrolyte, but boxed off from the outside world, so that the outside world only sees the two electrical connections; and investigating the object’s Blackbox behaviour, based on what happens when connecting an electrical circuit to those two connectors, we find the relationship: Q=C.V
So, here is the real answer to your question, as opposed to the purist one that I gave at the top of this reply:
Capacitors are designed in such a way that C remains (very nearly) constant with changing Q (and hence, V is (very nearly) linearly dependent on Q); whereas rechargeable battery cells are designed in such a way that V remains (very nearly) constant with changing Q.
(No-one ever talks about it, but the implication is that, for the rechargeable battery cell, the value of C not only changes, but is (very nearly) linearly dependent on Q. Note, also, that Q is measured in Coulombs, and that Ampere-Hours are just Coulombs multiplied by 3600.)
- Both the electrical components can generate a potential difference across another electrical component say resistor.
- They are used to store energy which can be tapped by connecting them to a closed circuit.
- The potential difference generated by a battery is constant, whereas, that generated by a charged capacitor changes exponentially with time.
- Though both the components are used to store energy, they achieve it in different ways is, battery stores electrical energy in the form of chemical energy, whereas, a capacitor stores it in the form of magnetic field.
- While charging a battery the polarity of the circuit must be reverse of that while using it, whereas, while charging a capacitor the polarity must be the same as it should be while using it.