The intent of any kind of tropical headgear is, of course, to keep the sun off the head and face and allow fresh air to circulate around the head, so that the user remains relatively cool. Caps and bush hats are good at keeping the sun off the head and face, but the heat of the sun is still transferred to the head by the fact that the cloth of the hat is next to the skin. Plus, these hats don't always do a good job of circulating the air underneath the hat because the ventilation is poor. By contrast, the pith helmet is designed to put a space between the hat and the head, to minimize heat transferrance and to allow air to freely circulate around the top and sides of the head. That's why the pith helmet headband is not flush with the sides of the helmet.
Most pith helmets have a wide brim that flares out to protect the face and sides of the head.
The only negative things about the pith helmet are that some of them are heavy, and none of them can be folded up and tucked into your luggage like a soft cap can. It has a classic appearance that lacks the exaggerated features of many of the other styles. The South African style from WW2 is hideous, as are most of the Italian ones. For one thing, the brim, in my opinion, is too broad, and looks a little silly when compared to the German and British styles. So to me, the US helmets are like a cheap version of the classic European styles.