The winglet's function is to break the vortex that forms at the tips of the wings. A vortex creates substantial extra drag. If you've heard of ground effect, where an airplane has less drag close to the ground, this is because the ground breaks the vortex. Winglets have this same effect, though require high structural strength at the weakest part of the wing.
"The winglet's function is to break the vortex that forms at the tips of the wings"
I believe I read that the effect would be the same if the wing were merely extended, but that winglets have the same effect without having to increase the horizontal wingspan......