brain damage vs trauma

brain damage and trauma both are nouns.

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

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

brain damage (noun)	trauma (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.	an emotional wound or shock often having long-lasting effects
	any physical damage to the body caused by violence or accident or fracture etc.; the condition of an injury