According to the infallible Wikipedia,
"Quartersawn boards have two advantages: they are more resistant against warping with changes in moisture and, while shrinkage can occur, it is less troublesome.
In high-end string instruments, the neck and fretboards can be made from quartersawn wood since they must remain stable throughout the life of the instrument, to keep the tone as invariable as possible. In acoustic guitars, quartersawn wood is also often used for the sides which must be steam bent to produce compound curves. This is partly for structural reasons, but also for the aesthetics of highly figured timbers being highlighted when sawn this way. On high-end electric guitars and bass guitars quartersawn wood is often used as the base material for the neck of the guitar, since this makes for a stronger and straighter neck which aids tuning and setup stability. In acoustic guitars, quartersawn wood is often used, especially for the sides which must be steam bent to produce compound curves. This is partly for structural reasons, but also for the aesthetics of highly figured maples being highlighted when sawn this way.
The second advantage of quartersawn wood is the decorative pattern on the board, although this depends on the timber species. Flat sawn wood (especially oak) will often display a prominent wavy grain (sometimes called a cathedral-window pattern) caused by the saw cutting at a tangent to a growth ring; since in quartersawn wood the saw cuts across the growth rings, the visible grain is much straighter; it is this evenness of the grain that gives quartersawn wood its greater stability."
Makes sense.....