I'm a Sony shooter with only a few lenses. I use to shoot Canon during the film days. I tested both the Canon 5D mkIII and the Nikon D800 and here are my results. I tested a 5DmkIII with a 24-70 f2.8 lens and a Nikon D800 (not D800e) with a 24-70 f2.8 lens.
High ISO: About the same, except D800 has a lot more detail to work with. In Lightroom, I can save a higher percentage of ISO 6400 shots because the D800 has more detail. Canon seems cleaner initially in Lightroom but when the picture fits into a 24" 1920x1080 monitor or a 64" Samsung plasma TV, the Nikon looks a tad cleaner, noise less noticeable. I think the Canon looks cleaner in Lightroom because it is just a smaller picture. But displaying ISO 6400 shots on a monitor or TV, Nikon looks nicer in general. Both cameras at ISO 12,800 look awful and not recoverable in Lightroom. It might look acceptable as a really small pic but why the heck would you buy an expensive camera to display crappy looking pictures?
Frames per second: Easy winner. Canon can shoot 6 FPS, Nikon 4 FPS. However, in practice I think 4 FPS is pretty good. None of these cameras are really Olympic style sports cameras.
Resolution: Easy winner. Nikon's RAW files are more detailed, almost 3D like. I can't really explain it other than the pictures look more real. I can crop a photo to 1/3 it's size (12 megapixels) and it still looks stunning. I wonder how much better the D800e version is. I'll have to wait until my friend receives his to find out.
JPEG: Easy winner. Out of the camera, the Canon JPEGs are phenomenal. The processing done is quite remarkable.
RAW: Easy winner. If you shoot RAW, Nikon is it. Also there is an issue with the Canon with the color red. I think the color is overblown at times because all the details are loss and not recoverable in Lightroom. Not always but it has happened at least twice. The same photo on the Nikon kept all the details.
Autofocus: About the same. Canon and Nikon have awesomely quick autofocus and I couldn't determine a difference. The only caveat is that Nikon focuses better in lowlight (without the autofocus assist lamp) and also the Nikon focuses when there is almost no light (with the autofocus assist lamp). Why the heck doesn't Canon include an autofocus assist lamp is beyond me. Also, Nikon's face detection is extremely useful because it focuses right on the eyeballs.
Flash: Easy winner. Canon doesn't have built in flash. Nikon flash worked surprisingly well.
Dynamic Range: Easy winner. Nikon knocked it out of the park. I got a lot less blown highlights with shots with white clothing and more realistic blue skys. Also, there is a lot more headroom on both the highlights and shadows on the Nikon when editing in Lightroom. How did Nikon have better shadows and highlights! They have to share some of that technology!
Video: Suprisingly about the same. I would have thought that Canon's lead in video would maintain. Surprisingly, the Nikon's video was just as awesome as the Canon. Nikon's video has a bit more detail and is definitely a little sharper than the Canon. I didn't test Nikon's uncompressed HDMI out, although it seems to be a useful feature (this is like RAW HDMI output for video). Canon should adopt uncompressed HDMI out also.
Price: Easy winner. Why anyone (who doesn't already have Nikon or Canon lenses) would buy this Canon for $500 more than the Nikon would need to think twice. I can see why the Nikon is selling so much better than the Canon, at least on Amazon.
Comfort: Both about the same weight. Both feel nice in the hand. Canon possibly slightly more comfortable if you have bigger hands. Nikon maybe more comfortable with smaller hands. Both are fine though.
USB transfer: Nikon wins with USB 3.0. Skipping the card reader altogether by just plugging in the camera to the computer is convenient. Also the transfer speed is much much faster than Canon's older USB 2.0. This saves a lot of time.
LCD Screen: Canon has slightly better screen in direct sunlight (LCD facing up towards the sun). Nikon is better when the LCD is not directly facing the sun. Nikon's screen is crisper and more 3D like.
Weatherproofing: My last day of shooting was in a light mist/drizzle. I was shooting both cameras again for about 20 minutes when the Canon 5D mkIII developed some fogging inside the viewfinder screen. I could not wipe it away as it seemed to be inside the camera. I could no longer take pictures normally without live view. Nikon didn't have this problem and I continued to shoot the rest of the day with the Nikon in the same wet conditions without issue. I had planned to shoot at least 2 weeks with both cameras so this was definitely a bummer.
Lenses: Both Canon and Nikon's 24-70 2.8 lenses are great. I would say the Canon 24-70 2.8 is just a tad faster on focus. Nikon is slightly sharper in the corners. Both Nikon and Canon seem to have a very comparable lens assortment (although my wallet won't be happy buying so many new lenses!)
Well, after using both cameras for about a week, I kept the Nikon D800 and returned my (possibly water damaged) Canon 5D mkIII. Both are phenomenal cameras but D800 has definitely outclassed the 5D mkIII in this round. Maybe Canon will come back strong with its next version. It definitely has some catching up to do.
Thanks for reading my review and I'll also post this on the Nikon D800 and Canon 5D mkIII review page. Best of luck to all you photogs and enjoy these phenomenal cameras!