I can't comment on the "tone wood" front. I'm still relatively new to bass (two years), but more importantly, I'm not entirely convinced there's a difference when it comes to non-acoustic/electric instruments. I'm more convinced that the strings and electronics have the greatest impact. Heresy to some, I know.
That being said, I will say that the body woods I like are the lightweight ones. I have two basswoods (Aerodyne & CV Jazz), an agathis (Squier P-Bass Special) and an alder (Blacktop Jazz). The Aerodyne and P-Bass Special are very light and, by extension, a joy to play. The CV and Blacktop Jazzes are weightier, but still manageable. I don't know if I could play a mega-set with them exclusively.
I love the look of rosewood and ebony fretboards, even ebonol. (Again, more heresy to some) No slam on maple boards, it's just a matter of esthetics to me. Although a maple board with black block inlays does look rather fetching.
But on a purely "constructivist" angle, it's the neck shape and size that's probably the most important factor for me. I prefer a Jazz's 1.5" width over the Precision's 1.65". Also, a tighter/curvier radius works best in my hands. I started with the Aerodyne (7.25"), got a couple more with the current Fender standardized 9.5" radius and then tried a flatter 12" board. It's not that the latter is unplayable for me. Bob knows, I tried for quite some time. But when I picked up the Aerodyne again, after many months of using the flatter neck, it felt so much more comfortable in ways I had never noticed before.
I'm curious how well I'd take to a compound radius fretboard. However, they seem to be pretty expensive, or at least, reserved for more expensive basses.