I have used the Nikon D800/D800E now for over a year, and a 6D since early February. I have shot the D600 but stayed away from it in the 4th quarter of 2012 due to excessive dust issues and returns (at one time my local dealer had 25 units stacked up to return to Nikon). Based on my brief shooting with the D600, I felt the results were pretty much identical to the D800/E except for the extra resolution. My photography is all landscape, no tethering, some wildlife. I found the D600 had the same amazing DR in the iso 100 range as the D800 possibly just a bit more. The higher iso range limit for my work was around 2500 iso with 3200 a push in ambient temps in the high 60's (F) as much higher than that I found the noise too destructive.
The Canon 6D was introduced to me on a night shoot in February, I was so impressed with the 6D I went out and purchased one the next day. The two things that impressed me immediately were the fact that the 6D had very clean higher iso's up to 6400 (about my max) and the noise in the normal range was like no other Canon noise in that it did not have the traditional green/red blotchy noise that for years I had struggled with forcing HDR type photography. You can't push the shadows as much with the 6D as you can with the D800/D600, but the noise to me is more like film grain, and less digital. It cleans up with much less image destruction.
The 6D outperforms the 5D MKIII in noise as I found the noise characteristics of the 5D MKIII to be the same as other older Canons. Red green banded noise that causes image destruction and required exposure bracketing to get an acceptable solution. On a regular basis I will pull up the shadows on D800 exposures as much as 2.5 stops with no noise or loss of color/sat in the shadows. For a landscape shooter the D800 DR was revolutionary to me and still amazes me.
My only real issues with the 6D are the fact that Canon should have just carried their tried and tested camera interface forward, i.e 5D MKII, MKIII, instead they IMO tried to figured out the worst possible combination of buttons, knobs and dials to give the worst possible user interface I have ever seen.
The user of the 6D has the ability to load the Magic Lantern firmware and unlock a HUGE amount of potential. If you use this firmware you will find yourself asking, why did Canon not do this. The firmware is free, a bit tricky to load ( I strongly recommend having someone who has it loaded help out) but it is a great asset to the 6D (5D MKIII 5D MKII 60D) also. I have been running it now for 1 month and have had no issues.
The 6D is clean in the upper iso range up to around 6400, much past that and the noise gets pretty harsh. However as stated before the noise at the higher iso ranges shows much less of the destructive red, green banding.
LIve view on the 6D is very straight forward and in lower light it's superior to Nikon as Canon somehow buffers out the noise. In low light Nikon shows a lot of noise and can get hard to focus with Live View. There already has been a lot written about Nikon's "over zoom past 100%) for me it's much easier to use when backed off 3 and you can set a preset for this.
The 6D works with Canon's standard plug for their intervoltmeter, with the D6600 you will have to purchase an adapter to use the Nikon MC-36 or one of it's ebay clones. There may now be a non Nikon model that gives you intervoltmeter functions with the port the D600 has. Yes both the D600 and D800 have an intervoltmeter built in, but they limit you to exposures of no longer than 30". If you are working at night you will need longer.
AF on the 6D is the same number of points as the older 5D MKII, however the AF to me seems a bit better in the ability to lock on to a subject the first time in lower light than the 5D MKII. The newer 70D has Canon's most advanced AF to date, and I have not read much about it's abilities yet. The D800 has 51 points, and most of the time they seem to do a good job, however in low contrast, low light situations, I have trouble getting focus lock. If you are using a telephoto lens this problem can get a bit worse. My experience is with the 200-400, a lens which I love, but in low light at 400mm F4, my D800 will miss about 45% or so of the time with the center point. Working small song birds from 15 to 20 feet with the D800 in mixed light can be a bit frustrating as the camera can't seem to lock and many misses are so close. Hands down in my experience the AF of the 5D MKIII is the best I have ever used in both AI and low light focus but this is beyond the scope of your question.
Nikon's D800 family lock you out of the camera when long exposure noise reduction is turned on, i.e. take a 2 minute exposure you have to wait another two minutes while the dark frame is written. Canon on the 6D buffers this operation to the background and frees up the camera. In fact the 6D can buffer this out very well, allowing a least 2 40 minute exposures with the dark frame running in the background. You will be locked out before you can take a 3rd exposure for about 20 minutes while the buffer is cleared. I was very surprised to see this with my D800 as it can really slow down workflow, especially if you are shooting a series of 10 second or longer exposures. I am assuming the D600 works the same way.
Both cameras will give you excellent results and both are an excellent value.