Differentiation

They used the snow to differentiate between the real events and events if George hadnt been born. George realizes hes back in the real world once he sees the snow has started again

The Butterfly Effect

Weather is a complex system, it is affected by virtually every object on the planet. Recall Malcolm explaining the Butterfly Effect in Jurassic Park: A Butterfly flaps its wings in Peking, you get rain in New York instead of Sunshine. So it's possible that all the differences caused by Georges absence could produce no snow on that day.

It wasn't George

Remember, Clarence asks his superior Joseph (or maybe a higher power) to give George what he has wished for by making it as if he had never been born. The snow stops when George's wish is granted. We can assume that the same power that made it so that George had never lived also changed the weather. This could have been done to help convince George that his wish really had been granted, or to make it a little easier for him to travel as well as to see the changed town.