breakup vs falling out

breakup and falling out both are nouns.

Word	Noun	Adjective	Verb	Adverb
breakup	Yes	No	No	No
falling out	Yes	No	No	No

As nouns, falling out is a hyponym of breakup; that is, falling out is a word with a more specific, narrower meaning than breakup:

Other hyponyms of breakup include breach, break, rift, rupture, severance, break.

breakup (noun)	falling out (noun)
the termination or disintegration of a relationship (between persons or nations)	a personal or social separation (as between opposing factions)
coming apart