Precise direction input. Especially useful when you need to input rapidly.
Easy answer is the Fighting genre. But the need is in other genres as well. The time it takes to tilt and release an analog stick is often too long, and will never be as good as a digital d-pad.
Role playing games are a big reason for me to keep the d-pad. Perhaps mainly for menu screens, when going through long item lists, d-pad means more precision in quickly getting to and stopping on what you want to select.
Movement in any 2D game is much better with a d-pad than an analog stick.
And this may sound odd, but I still prefer to play driving games like Gran Turismo 5, with the d-pad. Just feels like I have more precise control over my steering than I do when attempting to use an analog stick.
It sounds like you just want an Xbox 360 controller to be compatible with the Playstation. That is basically what you described.
As for third party, that would possibly save you money. So it'd be a win anyway, right?
--- --- ---
I don't intend to be rude to anyone, we all have opinions and preferences as to what feels right in our hands, we are all different. But something I see that keeps popping up is a desire for one company to make their stuff just like the other company because it is better etc.
What I want to say is, perhaps some of us are simply Xbox players, some Nintendo, and others are Playstation. Play the system with the hardware that is more suited for you, and accept that other people like the other console's controller the way it is.
Yes, there is always room for improvement, no controller is perfect. Not saying that. Just that, we still need variety in our gaming choices.
Not one single box of stagnation.
Touch Pad.