In response to the revolutionary Nintendo 64 controller, Sony retooled their own controller to make it more suitable for the future of 3D gaming.
Instead of adding one analog thumb stick like on the Nintendo 64's controller, Sony added two to theirs. I suspect this was mainly in order to maintain the controller's symmetry as there was no convenient location to add just one. The "analog" button was added to enable or disable the
thumb sticks - a way to retain compatibility with previous games.
Sony also added an innovation of their own in the form of a push-button within each
thumb stick. The sticks click when pushed in, essentially creating the equivalent of two additional shoulder buttons. Lastly, Sony incorporated two vibration motors within the controller's shell - adding a built-in equivalent to Nintendo 64's Rumble Pak.