Silent shooting mode normally (*please see the end of the answer for a notable exception) does not affect Image Quality in any way. Rather, it affects the way your 5D mk III cycles the mirror and shutter curtain for each shot you take.
When shooting via the viewfinder:
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. The mirror then drops back down and the shutter curtain reset, again at a slower speed than normal. [With some other Canon models after the second curtain closes 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 slower mirror movement also increases the "shutter lag', which is the time between the instant the shutter button is fully pressed and the shutter curtains begin to move, and increases "mirror blackout", which is the amount of time the viewfinder is dark while the mirror is not all the way down.
The silent mode continuous option is similar, but allows multi-shot bursts. The mirror swings up more slowly. After each exposure the mirror and shutter mechanisms will 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 Autofocus or Manual Focus:
Mode 1 begins with the first curtain open prior to the shot (so that the sensor is exposed to give a Live View image on the camera's LCD screen). When the shutter button is pressed the sensor is cleared and then electronically exposed from bottom to top (top to bottom of the inverted image) to begin capturing the image. The exposure is ended by the movement of the first curtain from the bottom of the light box back to the top of the light box. This is the same direction the first curtain is moved when it is reset following a still shot taken using the viewfinder. The second curtain never moves throughout the entire Live View - Silent Shooting cycle with mode 1. For each frame exposed, the only sound is the first curtain closing to end the exposure and then reopening for the next shot (if the shutter button is held down for a continuous burst) or to allow the sensor to send an image to the LCD screen for Live View (if the shutter button is released).
Mode 2 is a single shot mode. The shot begins just like Mode 1. But the second curtain closes from the top to the bottom of the light box (bottom to top of the inverted image) to end the exposure. Once the second curtain closes it doesn't reopen to expose the sensor for Live View until the full press of the shutter button is released. The first curtain never moves throughout the entire Live View - Silent Shooting cycle with mode 2. For each frame exposed, the only sound is the second curtain closing to end the exposure. Once the full press of the shutter button is ended the sound of the second curtain reopening to allow the sensor to send an image to the LCD screen for Live View will be heard.
Disable allows shooting with Canon's Tilt-Shift lenses or when using extension tubes (and presumably any third party lens that has an aperture manually set on the lens). When the shutter button is fully pressed using this setting the first curtain resets to the top of the light box, the sensor is cleared, and the first curtain reopens to begin the exposure. The exposure is ended by the conventional movement of the second curtain from the top to bottom of the light box, the sensor is read out, and the second curtain is reset to the top of the light box while the first curtain remains open in the bottom of the light box.
(*The exception would be when you are using shutter speeds subject to allowing the vibrations produced by the mirror and to a lesser extent the shutter curtains to affect the image. Although it varies based on camera design, most researches that have thoroughly tested such vibrations place shutter speeds between about 1/100 second and 1 second as the most vulnerable to mirror vibration affecting the image. Any shorter exposure time and the image has already been recorded by the time the vibrations reach the parts of the camera that affect the image. Any longer and the vibrations don't last long enough to have much effect in the same way that someone running rapidly across the field of view of a 30 second exposure won't show up in the image. Do note that you'll only ever be able to detect blur caused by mirror vibrations if the camera is mounted on solid footing such as a sturdy tripod, the shutter is released using a wired remote cable, an IR remote, or the self-timer and the subject is also perfectly static.)