hotel detective vs private eye

hotel detective and private eye both are nouns.

Word	Noun	Adjective	Verb	Adverb
hotel detective	Yes	No	No	No
private eye	Yes	No	No	No

As nouns, private eye is a hypernym of hotel detective; that is, private eye is a word with a broader meaning than hotel detective:

hotel detective: a private detective employed by a hotel or retail store
private eye: someone who can be employed as a detective to collect information and assist in criminal investigations

Other hypernyms of hotel detective include operative, PI, private detective, private investigator, shamus, sherlock.

hotel detective (noun)	private eye (noun)
a private detective employed by a hotel or retail store	someone who can be employed as a detective to collect information and assist in criminal investigations