Minecraft wiki entry for Redstone_circuits, associated text: Neither repeaters nor pistons 'burn out', unlike redstone torches.
Piston circuits are often (not always) smaller and/or faster than their redstone counterparts. This allows building devices such as fast clocks and "instant" signal transmission.
Pistons' ability to move blocks within the world makes them a natural for memory circuits, as well as the obvious doorways and switchable bridges. With slime or honey blocks involved, entire structures can "get up and move" (see also the Flying Machines tutorial).
Piston circuits can sharply reduce the use of redstone in favor of wood, stone, and iron.