Wool is in deed a fur from sheep but it is sheared. Fur felt is not sheared.
I think the big deal is that fur will "do the job" for decades but wool wears out much quicker, falling apart eventually.
Blasphemy alert!
I got nothing against wool felt hats. If what you want is a low-priced hat that you won't kick yourself over losing or sitting on, or a hat you wouldn't hesitate handing off to some girl who looks real cute in it, well, by all means, buy a cheap wool hat.
But don't pay something in excess of a hundred bucks for a 40 dollar hat. There are retailers now charging such sums for all-wool hats. And there are apparently enough people paying those prices for those retailers to keep on charging those amounts.
It's possible to make a high quality wool hat (I just posted one) or blend (yes there were beaver - wool blends). Some were at the highest price point (for example vicuña ).