I've listened to sound samples of basses where the only differences was the body wood, and ones where the only difference was the fingerboard wood, and I could hear differences.
Whether my mind made those up or those differences were actually there, that was enough to lead me to believe that wood does affect tone, enough to notice in a recording.
In a live situation? There are so many factors that affect tone that it's impossible to isolate. Live band tone is a crap shoot (not saying it can't be good, obviously it can be, but predicting what it will sound like by selecting the right woods for your bass is pointless compared to your amp, the room, the rest of the band, the sound equipment and operator, etc, etc...).