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1. Battery is a source of energy for a circuit, whereas capacitor is a passive element, which draws energy from the circuit, store and release it. 2. Usually battery is a DC component, whereas capacitor is mostly used for AC applications. It is used to block DC components in circuits.
You can make a very simple oscillator by connecting a capacitor and an inductor together. If you've read How Capacitors Work and How Inductors Work, you know that both capacitors and inductors store energy. A capacitor stores energy in the form of an electrostatic field, while an inductor uses a magnetic field.