A "hotel detective" was, and is, just a plainclothes security officer for the hotel security team. They are called "detectives" by analogy with the rank system many police departments use (or used to use), where there is a rank called "detective" below that of "sergeant", but above that of "corporal".

A hotel detective is not actively investigating incidents most of the time, or even at all -- he is a type of plainclothes security person who mostly is there to provide an invisible set of eyes and ears to the hotel security team. Most of the time he (or she) would be loitering around the exits or common areas, or occasionally roaming the hallways, just generally keeping an eye on things, but not standing out, or alarming guests in the way a uniformed officer might.