trophy case vs display case

trophy case and display case both are nouns.

Word	Noun	Adjective	Verb	Adverb
trophy case	Yes	No	No	No
display case	Yes	No	No	No

As nouns, display case is a hypernym of trophy case; that is, display case is a word with a broader meaning than trophy case:

trophy case: a case in which to display trophies
display case: a glass container used to store and display items in a shop or museum or home

Other hypernyms of trophy case include case, showcase, vitrine.

trophy case (noun)	display case (noun)
a case in which to display trophies	a glass container used to store and display items in a shop or museum or home