break someone's heart vs pain

break someone's heart is not a noun while pain is a noun.

Word	Noun	Adjective	Verb	Adverb
break someone's heart	No	No	Yes	No
pain	Yes	No	Yes	No

As verbs, pain is a hypernym of break someone's heart; that is, pain is a word with a broader meaning than break someone's heart:

Other hypernyms of break someone's heart include anguish, hurt.

break someone's heart (noun)	pain (noun)
	a somatic sensation of acute discomfort
	something or someone that causes trouble; a source of unhappiness
	emotional distress; a fundamental feeling that people try to avoid
	a symptom of some physical hurt or disorder

break someone's heart (verb)	pain (verb)
cause deep emotional pain and grief to somebody	cause bodily suffering to and make sick or indisposed
	cause emotional anguish or make miserable