Nikon D600 vs. Canon 6D in Low Light?

Nowhere near that. I've compared them.

In JPG there is a big difference if you compare them at "default" settings, but why would you do that (you wouldn't)? You need more NR and sharpening on the Nikon JPG to match the Canon. Overall I'd say Canon has a better JPG out of camera than Nikon (regardless of model).

In RAW, the difference is very small. About 1/3 a stop maybe, hard to say exactly. That is when taking both photos and comparing them at the same size. Noise is just very slightly lower on the 6D, however the D600 has more detail plus the photos just look better to me (which is why I now mostly use a Nikon). You can add NR to the D600 until it matches the 6D in noise or in detail. IMO when you do that with NR, detail will pretty much match and vise versa.

So overall IMO in terms of noise they are essentially equal but in some situations the D600 seems better to me, in RAW. If I shot in JPG (which I never do) I would most likely prefer the Canon.

That's IME with ISO noise on eahc camera. Other than that, each its own advantages and disadvantages, which vary depending on the photographer. I wouldn't mind using each of them.