The primary difference between a verb and an adverb is what they depict. A verb indicates an action being carried out by a noun. An adverb describes the nature, level, intensity or type of action. A sentence can have a verb that describes a noun's action, but an adverb can't exist without the presence of a verb.
In the statement, "Jill reads quietly," "Jill" is the noun or subject of the sentence, "reads" is the verb describing her action and "quietly" is the adverb that indicates the manner in which she completes the action of reading.