Capacitor and battery both perform the same function of storing and releasing an energy, however, there are essential differences between both of them due to how they function differently. Capacitors store energy in the form of an electric field while batteries store energy in the form of chemical energy.
The most important difference is that Capacitors are fabricated such that Capacitance C stays (just about) constant with changing charge Q (and thus, V linearly depends on Q); on the other hand, batteries are fabricated such that Voltage V stays (just about) constant with changing charge Q.
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In this article, the most important differences between Capacitor and Battery are elaborated on the basis of practical factors such as Energy Storage & Behavior, Life Expectancy, Voltage, Supply, Charging & Discharging Behavior, Types, Size, Composition, Charge Holding Capacity, and Applications.This following table covers the key Differences between Capacitor and Battery.
Difference between Capacitor and Battery
Characteristics Battery Capacitor
Energy Storage Energy is stored in a chemical form Energy is stored in an electrical form
Life Run for longer time Do not run for longer time
Energy behavior It is a source of energy in an electric circuit (acts as an active component) Passive element (stores and release energy)
Voltage Provides relatively constant voltage When it discharges, voltage decreases rapidly
Supply Provides DC Component It is usually used in AC applications and blocks DC Components
Types Alkaline, lithium-ion, lead acid, zinc-carbon Electrolytic, tantalum, plastic plate, ceramic, mica
Representation 
Charging & Discharging Slow Very quick
Size Small comparatively Large
Composition Metals and chemicals Thin metal sheets separated by an insulator
Charging Behavior Can be charged but will degrade with time Can be charged over and over without any significant degradation
Charge Holding Capacity Can hold charge for a longer time so can be used as primary energy storage medium Holds charge for a shorter time thus cannot be used as a primary energy storage medium
Application Use for emergency power, in robots & automobiles, local energy storage, in submarines Used extensively in pulsed applications for load sharing, in motor starters, as power back-up for most of the memory functions in various products