I'm not sure what the exact effect of this would be, but stormy weather, like hurricanes, are typically a part of the way the earth regulates temperatures and distributes heat built up in the tropical zones northward and southward into cooler climes.

So... if only the southern region continues to get more sunlight and the northern less, for one, you'd have less seasonality (only summer in the south, only winter in the north). Then, I'd think there's be no set "storm" season, but that massive storms would be a constant threat throughout the year. But I'm just conjecturing here...