brain damage vs hurt

brain damage and hurt both are nouns.

Word	Noun	Adjective	Verb	Adverb
brain damage	Yes	No	No	No
hurt	Yes	Yes	Yes	No

As nouns, hurt is a hypernym of brain damage; that is, hurt is a word with a broader meaning than brain damage:

brain damage: injury to the brain that impairs its functions (especially permanently); can be caused by trauma to the head, infection, hemorrhage, inadequate oxygen, genetic abnormality, etc.
hurt: any physical damage to the body caused by violence or accident or fracture etc.; the condition of an injury

Other hypernyms of brain damage include nervous disorder, neurological disease, neurological disorder, harm, injury, trauma.

brain damage (noun)	hurt (noun)
injury to the brain that impairs its functions (especially permanently); can be caused by trauma to the head, infection, hemorrhage, inadequate oxygen, genetic abnormality, etc.	the act of damaging something or someone
	a damage or loss
	feelings of mental or physical pain
	psychological suffering
	any physical damage to the body caused by violence or accident or fracture etc.; the condition of an injury

brain damage (adjective)	hurt (adjective)
	damaged; used of inanimate objects or their value
	suffering from physical injury especially that suffered in battle

brain damage (verb)	hurt (verb)
	give trouble or pain to
	be in pain
	cause damage or affect negatively
	hurt the feelings of
	cause emotional anguish or make miserable
	be the source of pain
	feel physical pain