The PS4 controller feels better. It is just more solid in the hands. I like the shape a lot better. What I don't like about the PS4 controller is that they moved the central buttons away from the center and more off to the sides. The result has been that, particularly during heated moments of gaming, I find myself hitting the Share button or Options button by mistake, especially when using the D-pad. I don't really want game-freezing buttons closer to the controls. Of course, I just got the PS4 recently, so I'm not entirely used to it yet. Hopefully, I will get used to it and stop it, but it has happened more than once.
For this reason alone, I would highly recommend that anyone who is used to a PS3 controller, but plans to use a PS4 controller in competition to get the PS4 controller early to practice with it.