In a nut shell
Batteries store energy with the use of two dissimilar metals, for example nickel and cadmium.
Capacitors on the other hand store energy through the separation of charged plates. Basically any two objects close together have capacitance (the ability to store energy). The spark when you get shocked by a door knob is kind of an example of energy stored by a capacitor.
You would need to take a Physicists II class to really begin to understand how a capacitor works.
Here's the difference in terms of "what can I do with it"
* far lower energy density (compared to battery)
* near instant charge and discharge
* manufacturable in nearly any voltage
* charge/discharge cycles measured in the 10s to 100s of thousands (and more)
* high energy density (compared to capacitor)
* limited charge/discharge depending on chemistry and design
* voltage determined by chemistry
* limited number of charge/discharge cycles (mostly chemistry dependent)
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