There are snaps on the brim allowing you to fold it up and out of your way allowing you versatility in how you wear your hat. Most have chin straps that allow you to hang the hat easily around your neck, and many times these straps are adjustable so you can make them work for the length you prefer.
The brim widths are the biggest difference. They will have a stiffer brim than that of the bucket variety. Boonie hat widths can vary although a wider rim is going to give you the best block from rays and rain.