I agree with the answer. The progressive (or continuous) is used to describe an event of happening either (1)taking place over some time or (2) intermittently over a period of time.
'By the time we get to Aomori' indicates the snowing should have begun before the people get to Aomori. Will be snowing therefore fits the situation better.
It will snow generally refers to the time when it begins to snow, and you can say, 'It will snow at lunch time tomorrow' and this means that the snowing will begin at lunch time. I'm disagreeing somewhat with xyontheodore.