As with any relatively unscientific field, there is a lot of lore out there that may have originally had a good scientific foundation, but the restrictions or specific conditions have long been forgotten and the answer takes on a life of its own out of context. The myth about eating snow seems to me to be one of these things.
The main point is that it takes a lot of energy, which when coming from your body means warmth and/or calories, to melt water to drinkable form. This is true, but the details are rarely considered. Russell has already give an good answer going over the numbers to show that the energy required is relatively minimal.
However, another point I never see addressed is that the energy is often free. If you're hiking up hill, even in winter usually, your body has to dump energy as heat. If you're cold and facing hypothermia, eating snow is probably not a good idea. Otherwise though if you're capable of dressing yourself so as to be too warm for whatever activity you are doing, you have excess heat that has to be gotten rid of somehow. Therefore except in unusual cases, melting snow with your body warmth is free since you just end up cooling yourself off less in other ways.
I have experienced this personally. Many years ago I was hiking up Mt Lassen in northern California in the summer. I had taken insufficient water with me and was getting thirsty. The weather was nice and I was hiking comfortably in shorts and a T shirt with a windbreaker in reserve in my pack. When I got high enough so there were patches of snow, I kept some in my mouth and got water from that. It definitely felt nice and made me less thirsty eventually.
The biggest problem you realize if you ever tried eating snow is that it takes a lot to get a reasonable amount of water. You simply can't put that much snow in your mouth at one time, and there is a lot of air in it, so the amount of water you get from a mouthful of snow is very limited. Still, it's better than nothing and makes you feel better. It does slow you down noticably since you have to pause to pick up fresh snow for another mouthful regularly. I suspect in warm conditions that you're barely replacing the water you are losing regularly anyway, but that's still better than losing it without replacing it.
So to me it seems that if you're thirsty and there is clean enough snow around, go ahead and "eat" it unless you are cold and hypothermia is a concern. Dehydration, after all, is a serious problem too.