It does make a huge difference with acoustic instruments ... for example, a violin, viola, and cello made out of the exact same materials will obviously sound different. Part of that difference is the size of their respective "empty space". (scale length and string length nowithstanding)
As far as solid bodies go, if you tap a large block of wood, and a small block of wood, you will hear different resonances. Strum the strings of a Steinberger and an Explorer, and acoustically they will sound different. However, as many have mentioned, once you add the electronics, the difference in tone is negigible (all other things being equal). Where you may notice the difference is in the amount of sustain. Prime example here would be to compare something like a Les Paul to an ES335. Which would sustain longer (without the help of mega-amp cranked tube saturated gain)?