Silent shooting mode does not affect Image Quality in any way. Rather, it affects the way your 5D mkIII cycles the mirror and shutter curtain for each shot you take.
The silent mode single option uses a slower speed to move the mirror up out of the light box. The first curtain opens, and then the second curtain closes as normal. But then, nothing else will happen until you release the full press of the shutter button. When you do, the mirror will drop back down, again at a reduced rate, and the shutter curtains will reset. This is very useful when shooting tennis or golf. You can expose a frame virtually silently at the height of a competitor's back swing and hold the shutter down until they have struck the ball and then release the button and allow the mechanisms to reset.
The silent mode continuous option is similar, but goes ahead and allows the mirror and shutter mechanisms to reset at a slower, and quieter rate. This reduces maximum frame rate from 6 fps to 3 fps.
In Live View the options are similar, but the mirror stays up as long as you are in Live View and are using Contrast Detection Auto Focus:
Mode 1 opens the first curtain prior to the shot and electronically exposes the sensor from top to bottom (bottom to top of the inverted image). For each frame exposed, the only sound is the second curtain closing and then resetting for the next shot.
Mode 2 is a single shot mode. The shot begins just like Mode 1. but once the second curtain closes it doesn't recycle until the full press of the shutter button is released.
Disable allows shooting with Canon's Tilt-Shift lenses or when using extension tubes.