Clear Skies Dragon Shout Refuses To Clear Winterhold

There are other things going on in Winterhold that could be affecting the weather. If you follow through the mage guild / college quest, you'll get an idea of what's troubling that region, and it's possible they'll elaborate further in DLC.

It's likely by design that you cannot use the shout there.

Actually, if you use the shout inside the Courtyard of the college, the weather clears and stays clear as long as you don't step out of college grounds... Weather turns back to snow almost as soon as you step back on the bridge to Winterhold...

Might be linked to the fact that the town was destroyed by the elements while the college hardly suffered. The most likely explanation is that the town is indeed cursed, and the college warded against the curse somehow, and the Thu'um is not strong enough to overcome the curse without the help of the college's wards.