I want to add a few things to the answers that are already given.
-If a detuned 4 string song works well on a non detuned 5 string depends highly on the difficulty of the specific song. If it is a simple song there is no problem. But if it is a technically difficult song with lots of open string riffs it will not translate well to 5 string so you are better of detuning.
-it also depends on how close you want to stay to the original. Do you want to play it exactly like the original bass player is doing it? Then detuning is better. Or transpose. Don’t you care? then use a five string.
-some songs are just meant to be in that tuning (especially open tuning!) and playing it on a 5 string doesn’t make much sense. Then skip the song or detune.
-If a detuned 4 string song works well on a non detuned 5 string depends highly on the difficulty of the specific song. If it is a simple song there is no problem. But if it is a technically difficult song with lots of open string riffs it will not translate well to 5 string so you are better of detuning.
-it also depends on how close you want to stay to the original. Do you want to play it exactly like the original bass player is doing it? Then detuning is better. Or transpose. Don’t you care? then use a five string.
-some songs are just meant to be in that tuning (especially open tuning!) and playing it on a 5 string doesn’t make much sense. Then skip the song or detune.