brain damage vs injury

brain damage and injury both are nouns.

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

As nouns, injury is a hypernym of brain damage; that is, injury 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.
injury: 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, hurt, trauma.

brain damage (noun)	injury (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.	wrongdoing that violates another's rights and is unjustly inflicted
	an act that causes someone or something to receive physical damage
	an accident that results in physical damage or hurt
	a casualty to military personnel resulting from combat
	any physical damage to the body caused by violence or accident or fracture etc.; the condition of an injury