Here on the west of Europe really cold air, and cold weather, tends to come to us from the east on an easterly wind. Quite often this air will have come from Siberia and so it will not have passed over much in the way of water to get here and as a result it will be fairly "dry" with little potential to produce snow.
Air that comes off the Atlantic however will tend to be warmer and to be carrying a lot more moisture and so it has considerable potential for snow if the conditions are cold enough.
I suspect this is where the saying has come from as really cold continental air is unlikely to give us snow. This also tends to be why snow doesn't lie for every long here as it often arrives on the leading edge of warmer air off the Atlantic. As others have said this warm, wet air meets cold air and causes snow but if the warm air keeps pushing east then the temperatures will start to rise and the newly fallen snow will melt.
Here in Northern Ireland we were on the warm side of the weather front that gave you snow in parts of England and so we've had no snow over the last while and, in fact, the conditions have been quite nice.