It's got very little to do with stiffness; in some woods there's a difference, some woods are stiffer in the flatsawn direction (which, incidentally, is the traditional grain orientation for violins and mandolins), some woods, it doesn't matter. What is difference is shrinkage and movement, although again, depends on the wood. With mahogany it doesn't matter much, for example.
Basically, quartersawn is about stability more than anything else. Should stay flat and not warp.
Figured stuff often is less stiff, less nice 'tap tone', often because its grain wiggles every which way. You get a lot of runout, few long, continuous fibres, so yes, material properties differ.
But it's nowhere near as simple as 'quartersawn' or 'flatsawn' being better tonally, or stiffer, or more stable. Depends on the species, depends on the piece.