The biggest difference between the 6D and its older sibling is price: right now you can pick up a 6D for nearly $1500 less than the 5D Mark III. Despite the price difference, the 6D maintains much of the 5D Mark III's functionality, even adding some features.
Let's get another big difference out of the way: autofocus points. The 6D offers a measly 11, just one of which cross-type, versus 61 and 41 offered by the 5D Mark III. AF points are critical to autofocus tracking, so you can eliminate the 6D from consideration if you plan to do a lot of sports photography.
The 5D Mark III is also the better camera if you do a lot of video work. We found the 5D Mark III to be highly effective at limiting video moiré, which wasn't the case for the 6D. The 5D Mark III's headphone jack is also invaluable for shooting video.
Beyond AF and video advantages, the 5D Mark III offers a number of features not found on the 6D: more rugged body, higher shutter life rating, more card slots, decreased shutter lag and faster continuous shooting, among others.
The 6D still manages to best the 5D Mark III in certain ways. By opting for the 6D, you gain built-in wifi and GPS, while losing 20% of the 5D Mark III's weight. You'll also enjoy a much quieter shutter.
What it comes down to is this: if you make your living off the camera, the extra $1,500 for the 5D Mark III is a no-brainer. On the other hand, if you don't plan to stress the AF system, don't do a lot of video or you prefer Wi-Fi and GPS to the grab bag of features offered by the 5D Mark III, pick up the 6D body-only or in a kit.
Superb still and video image quality; Powerful, fast, and accurate AF system with loads of cross-type points, loads of configurability and great frame coverage; Rugged, weather-sealed body with great control layout and user-interface configurability.
Dynamic range is limited by noise in deep shadows; Somewhat heavy-handed noise suppression and sharpening at default settings. (Shooting RAW avoids both.); No AF illuminator.
Excellent image quality on par to more expensive full-frame DSLRs (including the 5D Mark III); Responsive all-around performer; Superior HD video-shooting chops; Built-in Wi-Fi with remote control and sharing features, Built-in GPS and geotagging.
Lacks a built-in, pop-up flash; No external headphone jack; Rather basic 11-point autofocus system; Mediocre burst speed.