dispute vs fall out

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

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

As verbs, fall out is a hyponym of dispute; that is, fall out is a word with a more specific, narrower meaning than dispute:

Other hyponyms of dispute include brawl, wrangle, spat, polemicise, polemicize, polemise, polemize.

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

dispute (verb)	fall out (verb)
take exception to	come to pass
have a disagreement over something	have a breach in relations
	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