Generally considered the same and used interchangeably, breathing and respiration are actually very different from each other. Breathing is, in essence, the physical process of inhaling the oxygen and exhaling the carbon dioxide. Respiration is a chemical process by which we consume oxygen and produce carbon dioxide gas and energy molecules. While we can control the rate of breathing by taking deep breaths or shallow breaths or breathing faster (think of the differences between breathing during running and sleeping), we cannot control the rate of respiration. Respiration takes place at the cellular level and is a continuous process. It is a part of our metabolic cycle and is a catabolic process (meaning it involves the breakdown of food into simpler molecules along with the release of energy in the form of ATP molecules).
Hope this helps.