To a large extent, it depends on what you're doing with that flex/stiff board. In general, the faster a board goes the less stable it becomes. This can be countered by a) lengthening the board and b) stiffening the board.
When you're talking about hitting big kickers, a little stiffness in the board will help you take off cleanly and land with less reverberation coming through the board and your legs.
With jibbing surfaces, flexible boards allow you to "grab" the rail or box more easily. Try balancing a pencil on your finger and then try balancing and cable or rope on your finger. That's hugely exaggerated, but seeing the ends of a spectrum help us understand the shades inbetween.
When it comes to jumping or ollieing, you need to strike a delicate balance. Too soft and the board won't be able to convert potential energy into kinetic energy. This is where the pop comes from when you jump. Too stiff and you won't have enough strength to bend the board and transfer energy. Neither extreme results in an efficient ollie.