I have one of the first-generation "Seattle Sombrero" Gore-Tex hats... somewhere.

I'm not familiar with their recent hats, but that one was fairly unstructured and limp, with a low crown. I found it too hot in the rain, and too hot in the sun.

The old cowboy hats (especially the real ones, the old high-crowned "ten gallon" hats) protected your head from the sun by having airspace between your skull and the crown. This is one of the failings of most "boonie" style hats, the "crown" (just limp fabric, in a boonie) lies right against your skull, and so the full heat of the beating sun gets transmitted to your head.

Tilley hats take a different approach, they have a closed-cell foam panel in the top that acts as insulation (without having to look like Tom Mix or Hopalong Cassidy), and grommets (or now mesh) to allow air circulation from the sides. It works pretty well, helps the hat keep it's shape after crushing or packing (not just for cosmentics, but so that the crown has space for ventilation), and also means they float, which is nice for boating.

On the other hand, the more limp, unstructured hats are somewhat more packable.. though the Tilleys seem to do pretty well there. Matter of preference, I guess. They are much different.

Before I got my first Tilley, I had an "Ultimate Hat", which is a straight copy of the Tilley, and though it looked the same, and had a foam panel, it just didn't "work" as well, or hold it's shape as well.