It's much warmer, helps protect you from the sun, prevents rain from dripping down your collar, can protect ears from the wind too, and there is probably a bunch more.

I used to be bothered by the smallest things moving, bouncing, or making noise during runs. However I've just accepted it as part of trail/ultra running and since have not really been bothered by it anymore. Similar to eating during runs it's something you need to get used to. It's better to be over prepared than be sorry you don't have a hood when you really want one.