damage vs hurt

damage and hurt both are nouns.

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

As nouns, damage and hurt are synonyms defined as:

Other synonyms of damage include harm, scathe.

As nouns, hurt is a hyponym of damage; that is, hurt is a word with a more specific, narrower meaning than damage:

Other hyponyms of damage include detriment, deformation, distortion, ladder, ravel, run.

As verbs, hurt is a hyponym of damage; that is, hurt is a word with a more specific, narrower meaning than damage:

damage (noun)	hurt (noun)
the act of damaging something or someone	the act of damaging something or someone
any harm or injury resulting from a violation of a legal right	a damage or loss
the occurrence of a change for the worse	feelings of mental or physical pain
loss of military equipment	psychological suffering
the amount of money needed to purchase something	any physical damage to the body caused by violence or accident or fracture etc.; the condition of an injury

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

damage (verb)	hurt (verb)
inflict damage upon	give trouble or pain to
suffer or be susceptible to damage	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