Callus is simply a build up of dead skin. It presents as an area of hard skin and may become painful or un sightly to some people.
Heel Callus/Callused heels often develop cracks and may bleed. This is simple to treat and in most cases you are able to walk out without pain, and with normal looking heels.
Corns are a central build up of hard dead skin. Footwear is the most common cause, however causes do vary. Corns are often recurrent requiring removal every few months. This is usually painless and provides instant relief.
Corns may be mistaken for warts, a virus that can enter the skin. They require different treatments are contagious and can spread to the opposite foot and to other areas on the foot. There are many treatments for warts, most relatively pain free.