It am a BIG factor. Think about how violins would sound if they were made of styrofoam, or a guitar made of plastic.
Of the same wood, though:
The thinner the shell, the more low end.
The thicker, the more projection and high end!
The type of wood also affects the tone. I would really love to have a set that is all maple or birch shells. Maple has pronounced low end and high end. Birch has the same qualities with enhanced mids. The cheaper "mahogany" shells can be bright and unruley, but it depends on what type of "mahogany".