fall out vs dispute

fall out is not a noun while dispute is a noun.

Word	Noun	Adjective	Verb	Adverb
fall out	No	No	Yes	No
dispute	Yes	No	Yes	No

As verbs, dispute is a hypernym of fall out; that is, dispute is a word with a broader meaning than fall out:

Other hypernyms of fall out include altercate, argufy, quarrel, scrap.

fall out (noun)	dispute (noun)
	coming into conflict with
	a disagreement or argument about something important

fall out (verb)	dispute (verb)
come to pass	take exception to
have a breach in relations	have a disagreement over something
drop out
leave (a barracks) in order to take a place in a military formation, or leave a military formation
come as a logical consequence; follow logically