disagreement vs dispute

disagreement and dispute both are nouns.

Word	Noun	Adjective	Verb	Adverb
disagreement	Yes	No	No	No
dispute	Yes	No	Yes	No

As nouns, dispute is a hyponym of disagreement; that is, dispute is a word with a more specific, narrower meaning than disagreement:

disagreement (noun)	dispute (noun)
a difference between conflicting facts or claims or opinions	coming into conflict with
the speech act of disagreeing or arguing or disputing	a disagreement or argument about something important
a conflict of people's opinions or actions or characters

disagreement (verb)	dispute (verb)
	take exception to
	have a disagreement over something