This ^ All of it.
Your choice of pickups and EQ circuit will have the most profound effect on the sound of the bass.
I have an ash bodied, maple neck and board, corian nut and Schaller bridged bass, and another with a maple body, wenge neck and board, brass nut and two piece bridge. They have the same pickups and circuit in in pretty much the same position. Amplified, it's very very hard to tell them apart. Once the band starts up, it's pretty much impossible.
They feel, look and play differently, though a lot of that is down to their shapes, size, neck profile, etc.
The wood 'tone' is probably the least significant factor between them. I like Padauk.