DxoMark give the Canon EOS 6D a sensor rating of 82, while the Nikon D600 sensor gets a rating of 94. Sounds like a big difference. 12 points or about 14%. But what does that mean and why does the D600 get a higher score? It's worth noting that when discussing sensor performance we're really talking about the performance of the sensor and its associated electronics (readout circuitry, amplifiers etc.), since the two cannot be separated. Even wit a RAW file, the signal has to pass through multiple electronic stages after leaving the sensor before it is encoded into image data.
Well looking at the data, the biggest difference between the sensors is in dynamic range at low ISO. Here the EOS 6D scores around 12EV at ISO 100, while the D600 scores around 14EV. So what does this mean and why is 14EV better than 12EV. Well, for a well exposed shot of a normal scene, it doesn't mean a whole lot. Both a 12EV DR image and a 14EV DR image will look pretty much the same in a straight print. Where the extra dynamic range of the D600 would be an advantage would be in the recovery of a highly underexposed image, or where and image was processed in order to bring out maximum detail in deep shadows. In that case, shooting at ISO 100, you'd be able to pull a usable image out of a shot taken with the D600 with two stops less exposure than you could with an EOS 6D. This is at ISO 100. At ISO 400 the difference is closer to 1EV and by ISO 1600, the two cameras are equal. At higher ISO setting the 6D shows higher dynamic range and by the time you get to ISO 25600, the 6D has almost a 1EV advantage over the D600. 2EV is a significant difference since differences of only 0.5 EV are noticeable.
Looking at the noise in mid tones, the Canon EOS 6D and Nikon D600 are pretty much the same. The D600 has slight advantage at ISO 100, but the 6D has a significant advantage at ISO 25600, and over the middle ISO range, they track each other pretty well. The same pattern repeats with tonal range with a slight advantage to the D600 until you get to ISO 25600, where the 6D is again the winner.
Color sensitivity also follows the same path, though in this case the difference is a little larger, with the D600 leading the 6D by about 1 bit, and the order reversing at ISO 25600 where the 6D leads the D600. DXoMark themselves comment that differeces of 1 bit are barely noticeable.
It's worth noting here that the 6D sensor has pretty much the same performance as that in the EOS 5D MkIII, though it does seem to show slightly higher dynamic range.
I think the bottom line is that most users won't see much difference between the output from the 6D sensor and D600 sensor most of the time. However if you want to dig deep into the shadows at low ISO settings, the D600 sensor is certainly better. If you want to see the difference, I did directly compare the D800 sensor with the EOS 5D MkIII sensor in an earlier article, and the dynamic range difference between those two sensors is similar to what you'd expect between the 6D and D600 sensors.
So which is better, the 6D or the D600? Well, you didn't expect me to give you an answer to that, did you? As always "it depends". Clearly if you need very high ISO settings and want GPS and WiFi, the EOS 6D is your best choice. On the other hand if your most important needs are low ISO dynamic range and lowest cost, then the D600 is your best choice. As always there is no one absolute "best" camera for all photographers. For those with Canon lenses, the EOS 6D makes the most sense while for those with Nikon lenses, the D600 makes the most sense.
If forced to make a decision about which camera to give to a photographer who had no investment in either system, I'd probably go with the Nikon D600, mainly because of the current deal where you can get:
Otherwise, in the absence of any special deals and rebates, where both camera bodies have an MSRP of around $2100, I'd probably opt for the Canon EOS 6D.