A battery generates a voltage by a chemical reaction. There is a class of chemical reactions called redox reactions that involve the transport of electrons, and you can use the reaction to drive electrons through an external circuit. This is the basis of a battery. The battery will continue to provide power until all the reagents have been used up and the reaction stops. A battery generates a potential difference that is related to the free energy change of the reaction occurring in the battery. Note that there is no charge separation in a battery until you connect it to something and current flows.

A capacitor is completely different. It has a potential only because charge has been stored on it, and when you connect the capacitor to an external circuit a current only flows until all the charge has drained. Unlike a battery, the voltage on a capacitor is variable and is proportional to the amount of charge stored on it.

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

A capacitor is a device for storing charge. It is usually made up of two plates separated by a thin insulating material known as the dielectric. One plate of the capacitor is positively charged, while the other has negative charge.

The charge stored in a capacitor is proportional to the potential difference between the two plates. For a capacitor with charge Q on the positive plate and -Q on the negative plate, the charge is proportional to the potential:

If C is the capacitance, Q = CV

The capacitance is a measure of the amount of charge a capacitor can store; this is determined by the capacitor geometry and by the kind of dielectric between the plates. For a parallel plate capacitor made up of two plates of area A and separated by a distance d, with no dielectric material, the capacitance is given by :