I too like the sunrunner, with the French Foreign legion (or Ferengi?) neck flap attached.
A couple of reasons I haven't heard mentioned already:
On a longer trip where one goes into towns for resupply, I like being able to remove the neck flap and turn it into a more conventional looking baseball hat. Yes, I shouldn't concern myself with "looks", but my theory is that if I can clean up a bit and look less like a hobo, I might get treated a little better (?) and perhaps it will be easier to hitch a ride that way. Who knows for sure.
Another advantage over a wide brimmed hat is that a coat or poncho hood fits comfortably over it when there's wind-driven rain (or sleet or snow).
I've logged a lot of miles with my OR sunrunner; I actually bought a second one when the fabric of the first started getting thin/worn enough that I was getting tanned on my bald head under the hat. That took a good deal of miles & time, however, I don't mean to say that durability is a problem in normal use. I'm just saying that after quite a lot of use I'm still a fan.
Oh, one other thing it has over some types of wide-brimmed hat --- it can be put into a mode where you essentially have a chinstrap so that it won't blow off in high wind. In that mode, just the neck flap alone helps a lot when cold wind is blowing by your ears. And then it can be changed to wear it in a more "easy on, easy off" mode.