A particular note, say an "A" at 440hz, sound vastly different when played on a flute vs a clarinet. This difference is what is referred to as "timbre", and is a result of different harmonic overtones generated by the construction/physics of the different instruments. So when you pluck a certain note on a bass guitar, the waveform generated by the vibrating string is very complex and rich with harmonic overtones.
Now here's where the wood comes in: various woods interact with the harmonic overtones of the note in different ways - effectively cancelling out or reinforcing certain spectral elements (those overtones) as the string vibrates. All other things being equal (string type, pickups, etc, etc) different body woods will cause the string to actually vibrate in slightly different ways which of course affects the timbre.
Since the string is physically vibrating differently depending on woods, then it follows that the pickups amplify a different sound. This difference in sound quality is really timbre, not tone - though everyone uses that term.
That's the dollar tour anyway!
Hope that helps!
-M