If I have a departure before civilized people get out of bed, then I'll stay at the airport hotel (this usually meant the ORD Hilton when I lived in that area).

In the days I did mileage runs, and just needed a convenient place for the night, I'd stay at the airport hotel if it were convenient (TPA and MCO come to mind).

If I'm traveling out of the UK on Christmas Day and can't be sure any public transportation will be running at all, I'll stay at the airport hotel (did this once at Bewley's MAN).

If the airline is paying for the hotel because of the way connections are scheduled (NW GUM-NRT-ORD/SFO back in the mid-80s, arriving about 0600 and departing roughly 1600 or 2000) then I'll stay where they put me (happened to have an excellent view of the ramp -- that CO 72S looked puny among all the widebodies!).

Other than that, I think the only reason I've purposely stayed at an airport hotel was because it had a good reputation for plane spotting (Renaissance at ATL, I forget which at LHR).