Well, if you make a list of the items that influence the tone in order from greatest to least (rough estimate for order, anyway):
Pickups
Neck scale
Electronics
Neck wood
Guitar cable
Bridge/nut type & material
Strings
Body wood
Frets
When playing acoustically, the body wood will make a huge difference because you are also hearing the resonance in the wood. Plug it into an amp and the pickups only pickup the vibrating string. The wood has a slight affect in that the vibrating wood will have a feedback effect on the vibrating string. This is why the neck wood has a greater affect than the body wood. The neck can vibrate more freely than the body.
The guitar cable in conjunction with the pots and amp input creates a low pass filter which has a much greater affect than the body wood.
Changing from a 250k pot to a 500k pot can increase the resonance peak of a pickup by about 3dB.
Yes, the wood does influence the tone. I never said it didn't. Just compared to the other items listed above, it's small potatoes.
Now if you are putting some Bill Lawerence XL500XL pickups in a maple body with a maple neck, 1M pot, brass nut, tele bridge with a low capacitance guitar cable, then you might as well have a dog whistle. Then the only change you did was make the body mahogany, you probably wouldn't notice a difference.