cut out vs die out

cut out is an adjective but die out is not an adjective.

Word	Noun	Adjective	Verb	Adverb
cut out	No	Yes	Yes	No
die out	No	No	Yes	No

As verbs, die out is a hyponym of cut out; that is, die out is a word with a more specific, narrower meaning than cut out:

Other hyponyms of cut out include gouge out, rabbet, die.

cut out (adjective)	die out (adjective)
having been cut out

cut out (verb)	die out (verb)
delete or remove	become extinct
cease operating	cut or shape with a die
intercept (a player)
strike or cancel by or as if by rubbing or crossing out
form and create by cutting out
cut off and stop