As an example of the sonic differences that can occur with different woods, I compared three MTD 535-24 basses that were in stock at the Bass Exchange in LA when I was there recently, all of which had different neck/fingerboard/body combinations (plugging all into the same amp with no change in settings). Electronics were the same on all three and the strings were in similar condition.
In short, it was shocking how much difference there was. The overall tone of all three basses was similar, but there was quite a bit of variation in the amount of sustain and the way the instruments responded to different types of attack (pick, slapping, etc.). More surprising to me, the character of the treble, midrange and bass on all three varied noticeably in different ways.
The first bass had a maple neck/fingerboard and a swamp ash body. It had almost brittle highs, scooped mids and tight bottom end, not a great amount of sustain but a lot of punch. Probably good for modern slap-style playing. The second bass had a myrtle top, makore body, maple neck and rosewood fingerboard, and it was more sustaining, more piano-like and complex tonally with a broader and warmer bottom end. I felt like that bass was the one for me, and that I could get everything out of it that I would need to. The third bass, with a poplar body/maple neck/grenadillo fingerboard combo, was somewhere in between, with a bit more midrange and less sustain than the second one.
Having said all that, I was mostly window shopping at that point, but it was a very educational experience.