All three concepts—batteries, capacitors and flywheels—convert energy, store it and deliver power—but they're not created equal. Prompted by recent race-car designs as well as basic research appearing in my weekly Science magazine, I thought I'd do a bit of a study on my own.
A battery stores its electrical energy chemically. In charging a battery or releasing its energy through discharging, ions pass through its electrolyte from electrode to electrode. Lead/acid batteries are the traditional type, still evolving through improvements such as AGM (absorbed glass mat) technology. Nickel/metal hydride batteries made the hybrid a viable concept (as in Toyota's Synergy). The latest technology is lithium-ion, as exemplified in PHEVs (plug-in hybrids) and BEVs (battery electric vehicles).
Batteries are good at storing energy. But because of their chemical processes, they're limited in how quickly they can convert it. That is, they are limited in power (power being the rate of doing work, of converting energy over time).
In contrast to a battery's chemical storage, a capacitor stores its energy electrochemically on the surfaces of its electrodes. Because of this surface phenomenon a capacitor can quickly absorb or deliver its energy; that is, it's a high-power device. However, because only the surfaces of its electrodes are handling the charge, it has limited energy storage.
As described in Science, Vol. 335, 16 March 2012, researchers are working on this shortcoming by replacing conventional carbon electrodes with those of porous graphene. The latter is highly conductive—and has lots of surface area.
To summarize—and generalize—batteries are strong on chemical energy but not power. Capacitors (in their advanced form, ultracapacitors) handle high power electrochemically but thus far lack energy. Also, note that each of these can be engineered to trade off some benefits and shortcomings. Last, concerning electric vehicle applications, I note that energy dictates range; power dictates performance, both acceleration and efficiency of recharge.
The flywheel concept is utterly different from either, in that it adds a purely mechanical aspect. A flywheel stores energy in its physical spin, typically on extremely efficient bearings in a vacuum. It produces power on demand by converting the energy of this spin.
Clutches and gears can do this purely mechanically (the original Flybrid was such a device; see "Trackside: KERS Update," January 2009). However, the latest variation is a flywheel/electric hybrid, the spinning flywheel acting dually as a extremely high-speed motor/generator. In a sense, it stores its energy mechanically, but accepts and returns this spin energy electrically.
The Audi R18 e-tron Le Mans car uses this flywheel/electric approach, as does the Porsche 911 GT3 R Hybrid. The Toyota TS030 Hybrid Le Mans car uses ultracapacitors. As I'm writing this prior to their Le Mans battle, I'll not be betting on which is better. I note, though, that neither uses a battery, apart from buffering and conventional "hotel loads."