Calluses are a normal thickening of the skin of the foot in response to generalised pressure on the skin. If they are not uncomfortable they are best left alone as they do offer some protection. If painful they can be removed painlessly with a sharp scalpel by a Podiatrist.
Corns are a very localised thickening of the skin on the foot, often on the plantar, or bottom, surface, of the foot. They form a small hard lump which digs into the soft tissues when pressure is applied, often causing significant pain and limiting weight bearing activity. They can be caused by tight shoes, bone spurs under the skin, or high peak pressure in an area of the foot due to abnormal biomechanics in walking. Like calluses they can be removed with a sharp scalpel.
If they are persistent, corns can often be prevented from coming back by changing footwear or by wearing a specialised orthotic insole inside the shoe designed to offload the high pressure area.