Got to use and handle one today for the first time. There is a good reason why your D800 is heavier - it has the magnesium frame, whereas I found out the D600 is all plastic. Yes, it is lighter - I think it is lighter than my D7000 even, or at least it felt that way to me.
My major concern with the D600 is how light weight and "delicate" it feels to me. I think some people will love it for just that reason, but for me, I can be rough on my gear. I go into rough places. I once broke a part off my 4x5 sliding down a cliff into a river. Got a cut on my head too.
Point is, I sometimes go deep and hard into the bush and many other rough places, and not always nice weather either, and I would be constantly worried about the D600 is those enviroments.
So for a guy like me, the D600 would not stand up and take the punishment. Your mileage may vary. Bear in mind I think the image quality of the D600 is superb, so this is not the point. It is the physical durability of the camera that is to me, potentially lacking.
I am kinda disappointed that all the D600 reviews I have read, everything from magazines to different forum posts, I've sen very little reference to this point. One reason I went with Nikon years ago is their gear, for the most part, took punishment better than many other camera brands.
So I think it depend on when and what kind of enviroments you plan on using it for. My best advice is get to a camera store, take you D800 with you, and compare the two directly side by side and see for yourself what you think.