the blood vessels in the nasal cavity dry out during the cold weather causing irritation/burn sensation that precedes the sneeze. Oh wait, that's HOW we sneeze during the cold weather, but that's not the purpose, that's not the why.
Why we sneeze does have a lot to do with the irritation. That irritation signals a sneeze, the sneeze is initiated for the purpose of releasing stored up mucous to the irritated areas in order to coat them, protect them, and serve as a shield/barrier against further irritation/burning and potential infection. It's a immune system defense mechanism.