What is the purpose of Quick Mode (found on some Canon DSLRs) focusing in live view? To me it seems like a bad blend between live view and viewfinder mode that really doesn't cut it.
Of course it's a bit faster than contrast detection, but why not use the viewfinder instead if focusing speed is important. And if you're in live view mode and a good focus lock is required you're better of using contrast detection or manual focus.
It is quicker as the name implies and as you figured out. "Why not use the viewfinder instead if focusing speed is important?" Because you cannot. :) At least for me, this happens mostly when I shot with my dSLR above the head (it happens in photo journalism) and hence in order to frame I use the Live View. Also, when you shot video you must be in Live View. Ok, you will cut in post the small piece where the AF is hunting/working but you have changed the focus very quickly and have a new piece of movie focused successfully elsewhere.
Low-light. Well, this is big one. There's really no comparison between the performance of PDAF (especially cross and double-cross AF points) and CDAF in low-light / low-contrast situations. And when we say 'low-light' we mean almost any indoors environment.
Flexibility. This is true especially if you have high-end AF sub-systems like in 5D3 / 1DX where you can quickly change the way in which /and the place where the dSLR focuses. That box of CDAF is quite lazy and doesn't offer area expansion / small area focusing and other such things.
But of course, if you and your subject have the time and the patience then it is better to use Manual Focus in Live View.