schism vs falling out

schism and falling out both are nouns.

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

As nouns, falling out is a hypernym of schism; that is, falling out is a word with a broader meaning than schism:

schism: the formal separation of a church into two churches or the withdrawal of one group over doctrinal differences
falling out: a personal or social separation (as between opposing factions)

Other hypernyms of schism include breach, break, rift, rupture, severance.

schism (noun)	falling out (noun)
division of a group into opposing factions	a personal or social separation (as between opposing factions)
the formal separation of a church into two churches or the withdrawal of one group over doctrinal differences