Flex printed circuits are used because rigid printed circuit boards don't bend. Really.
A common application example is in a camera. There are buttons on all sides, controls to the shutter and lens, battery and sd card connections, an LCD, and all of that has to fit into an oddly shaped enclosure that fits the hand easily. You could do it with a bunch of tiny boards plus a lot of wires and connectors, but connectors are a common point of failure, they take up board space, the assembly costs would be high, and the whole thing would be difficult to test. It's much easier to create all of those small boards and connections as a single flat flexible board, which can be assembled and tested as a single unit, then be folded as necessary to fit in the enclosure with no connectors or wires needed. I should mention that most boards like this are actually rigid-flex, where portions of the board where components are mounted are rigid, and the parts interconnecting the boards are flexible.
