What is it?
At one level, it's all blindingly obvious. We are human and we sweat, especially when something is attached to the top of our head for several hours. So every hat requires something to absorb the sweat, stop it running into the eyes, protect the material of the hat and so on. Certain materials are appropriate for this function and the best of all is soft leather.
All men's felt hats should include a proper leather sweatband.
Unfortunately, however, during post-WW2 times of economic hardship, some hat manufacturers started to cut corners with sweatbands. The hats still looked pretty good. But they weren't as comfortable anymore, because the sweatbands were made of thick paper, painted to look like leather. Buyers of vintage hats must beware!