In common usage there is no difference between "breathing" and "respiration." Both terms are used to describe the process of bringing air into the lungs, exchanging oxygen and carbon dioxide and expelling the stale air. For example, the medical profession speaks of "respiration rate" to describe the number of inspiration cycles per unit time that a patient has.
Getting a bit more specific, respiration is actually a broader category than breathing, since there are methods of gaseous exchange that do not require breathing. Insects, for example, do not breathe, but do respire.
Breathing may be considered the physical action your whole body goes through to begin the process of respiration. Respiration is the action of removing air from the atmosphere, replacing it with the carbon dioxide in the blood stream, and moving that back into the atmosphere.
There is also a process called "cellular respiration" which is very different from the breathing/respiration of complex organism. Cellular respiration is the means by which individual cells use oxygen from the blood stream in combination with a sugar to release energy.
Cellular Respiration: The chemical reaction that occurs within cells to produce energy. It is essential for bodily function as it provides energy. The chemical equation for cellular respiration is:
Glucose + Oxygen => Carbon Dioxide + water + energy.
(Sugar) + Oxygen => Energy (Carbon Dioxide and water)
The scientific equation is:
Breathing or bodily respiration: The mechanical process that allows air to be transported into the lungs for the diffusion of oxygen into the bloodstream and diffusion of carbon dioxide into lungs which is then transported out.
(BREATHING IS AN EXCHANGE OF GASES)