I've read that capacitors can be used as batteries – does that mean they can be used interchangeably?
Is there any advantage to using a capacitor as a battery or in place of a battery?
Capacitors can indeed be used to store small amounts of energy. However, compared to a battery they have very low energy densities. As for being interchangeable with batteries, no not really. i.e. if you have a device that uses AA batteries, you will not be able to obtain a AA sized capacitor that you could simply place in that device. Even if you could, the AA batteries will last many hours of use, whereas a capacitor that size may only be seconds (at best minutes).
If however you are thinking about some sort of device that needs a few seconds to shut down gracefully in the event of a power failure, then a capacitor may be a better choice than a battery. The battery has a finite shelf life, whereas the capacitor will last indefinitely and there is no chance of a chemical spill, such as occurs with an aging battery.
Over the useful life of a battery, its voltage will remain relatively constant as it discharges, within a few tenths of a volt (above or below its nominal voltage). For a rechargeable battery, it is a slightly more pronounced at the end of its discharge cycle. Unlike the battery, the voltage of a capacitor is dependent on its level of charge, i.e. from q = CV we have V = q/C. When the charge (q) is high the voltage (V) is high and when the charge is low the voltage is low.
Having said all of that, capacitors serve many useful purposes other than storing energy, uses that cannot be performed by a battery.