What is the difference between respiration and breathing?

Respiration is the biochemical breakdown of glucose in the cell to prouce energy in a usable form of ATP (the body's energy currency). Respiration can be in the presence of oxygen (aerobic) or without oxygen (anaerobic), although there is the exception of cancer cells which preferentially choose to perform aerobic glycolysis and recycle lactate in the Cori cycle to provide their energy for proliferation. 

Breathing (some may say ventilation) is the mechanical process whereby the rise and fal of the chest and difference in atmospheric and internal body pressures causes the lungs to fill with air. Gaseous exchange then occurs at the surface of the alveoli to oxygenate the blood.