The Box has more than the stick. More internal memory, better graphics, better wifi with longer range, and a remote with voice control (voice only works with Amazon content not other apps like Netflix, a this time voice is pretty worthless in my opinion.) The new box can also do 4K / UHD if you have a 4K/ UHD TV.
That being said... I think for most people that the box isn't worth the extra $60. The box might be worth it for a tv that is really far from your wifi box, or for people that want to play Amazon/ android games. I have both and I do notice that the box is slightly faster than loading menus / movies but not that much faster. I think for 80% the $40 stick is the way to go.