Why should I choose 5D mk II and not Nikon D600?

If you will use it for video, I recomend Canon. Even the 5d Mark II is better at video then any Nikon camera.

For stills think also about what you might use the camera for and what lenses you will need. The Canon and Nikon lens line-up is very different and both have their strong contenders. Make a setup of lens and make a budget and see in time which brand will be easy to grow on.

A newer body is a better idea normaly so if you compare the Nikon D600, do it with the Canon 6D. It's price is going down and will get much cheaper soon ( the 5d Mk 3 just did also).

Don't be fooled by the Nikon increased DR. I strongly believe that you can get the same pictures with many of today cameras, the main difference is in the algorithm applied when reading the sensor and even more when converting to Jpeg. The increased DR in Nikon comes at a price - higher noise. So if you take a Canon RAW image and pull the shadows and highlights you will get very close to the Nikon DR.

Also you could try to rent both bodies to see which you like better. I chose Canon mostly because I like it better and as I said I feel you can get great images with any camera brand. It is important to feel good when shooting it. I would get frustrated with Nikon's menu every time I would use one. Also I feel the lenses I use are better in Canon version.

Good luck with your choice.