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Bass amplifiers are really similar to guitar amplifiers, with a few differences to improve bass response. Usually, a bass amp has a Larger speaker or more speakers, and a larger amplifier.
Larger/multiple speakers are usually better at reproducing lower end frequencies, and handling the much larger amount of power bass frequencies requires, usually having stronger built cabinets as well.
Bass amplifiers are less likely to have effects built in, and often include equalizers to modify the sound more than the standard treble/mids/bass on most amplifiers, as low end needs to be dialled in a lot more in live settings.