I am interested in knowing what the difference between the two features is. I don't shoot much using live view since I prefer looking through the viewfinder, but I would be grateful if someone helps me understand the differences for when I do occasionally use Live View.
There are two different types of Autofocus in most DSLR's. There is PDAF (or phase detect autofocus) and CDAF (contrast detect autofocus). PDAF uses a special focus sensor to measure the phase of the light coming from different parts of the lens (which is much faster) while CDAF uses the actual image sensor to look for the highest contrast it can get (which requires moving past the point of ideal focus and then coming back to it, which makes it slower.)
While some recent DSLRs have come up with ways to include PDAF sensors on the main imaging sensor, many models still use a completely separate sensor (either for cost or performance reasons). When you are using Live View, the mirror is flipped up and thus no light is going to the PDAF sensor or the view finder.
Normal live view focusing will use CDAF and hunt for the proper focus, which can have an advantage in certain low light situations and be more accurate in some cases, but it is also much slower. (Things like face detect are also possible depending on your camera model.)
Quick focus on the other hand will flip down the mirror so that the PDAF sensor can be used, make the necessary focus adjustments and then snap the mirror back up to take the photo. It is important to note that the image doesn't actually capture when you hear the first mirror movement, but rather after the second mirror movement.
To quote from your camera's manual (page 118):
"The dedicated AF sensor is used to focus in the One-Shot AF mode (p.66), using the same AF method as with viewfinder shooting. Although you can focus the target area quickly, the Live View image will be interrupted momentarily during the AF operation."
More specifically, what it does is to briefly put the mirror back into the optical path, thus enabling use of the normal PDAF sensor for focusing; hence you get much quicker focusing than with CDAF, but as noted there will be a brief interruption in the live view display while the mirror is in the optical path.