Most felt hats out there are made from WOOL felt instead of FUR felt. Simple economics. True FUR felt gets expensive - fast. But it does work better and last longer. You just have to pay more for it. So WOOL felt tends to be the common choice, and most available.
Felt hats work well for simple head protection. You won't get much insulation value from them, but you do get the shade and some rain protection (unless really treated with waterproofing).
In cold weather I generally wear a wool stocking cap/hat. And lately I've been wear my Tapaboard more often. It is pretty much a Medieval wool skull-cap with a wool skirt around the back that can be rolled down and then tied under the chin. When not needed that skirt rolls up to become something of a hat band. But they kind of went out of fashion in the early/mid 1700's. A VARY WARM wool cap/hat to wear when shoveling snow and the wind is blowing! That skirt hanging down and tied under the chin stops a lot of wind/cold. It does kind of look like those old French Foreign Legion hats when the skirt is rolled down.
During the summer heat, I wear a wide-brimmed straw hat. I get shade and some protection from the heat. When it rains, the straw starts to soak it up, expand a bit, and then becomes somewhat waterproof.
I personally have never liked leather hats. But I have had little experience with them. Those I did use tended to be too hot, too cold, and a soggy mess when they got wet. I have wanted to try wearing the old leather skull-cap style hats - like tank drivers or pilots in WW II. I'll have to sew one up one of these days.
Mikey - that grumpy ol' German blacksmith out in the Hinterlands