Patients are unable to eat or drink easily. This leads to owners taking the cone off and sometimes forgetting to put it back on.
Unfortunately, animals lick their wounds. This leads to infection and self mutilation and ultimately failure to recover. Many times this means more money to fix things that were not broken when the animal came in.
Cones become disgustingly caked up with food, saliva, dirt and whatever else gets trapped in the bulldozer like structure we place over their heads.
Dogs get cones stuck into the ground when they are being walked, this makes it almost impossible for them to sniff around to find the right spot to go to the bathroom.