Respiration reffers to intake of oxygen and its utilisation by cells for breakdown of glucose MOLECULES.. Carbon dioxide and water are released as by product along with ATP molecules...

However BREATHING is simply a phenomenon which reffers to intake of air rich in oxygen and exhalation of air rich in carbon dioxide..

Cell respiration causes a controlled release of energy from glucose. The released energy is used to make ATP from ADP.

Aerobic respiration..
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The first stage of cell respiration is called glycolysis. In glycolysis, there is an initial expenditure of ATP. Then glucose is broken down (in the cytoplasm) into pyruvic ACID (pyruvate) with a SMALL 'profit' of ATP.
The pyruvate produced by glycolysis contains a significant amount of energy. If oxygen is present then pyruvate is metabolized by aerobic respiration to yield a LARGE amount of ATP.
In aerobic respiration, pyruvate is transported to the mitochondria where the Krebs cycle and the electron transport chain produce CO2, water and a large yield of ATP.
A mitochondrion has an outer membrane that is smooth and an inner membrane that has infoldings called cristae. The cristae increase surface area of the inner membrane, thereby enhancing the productivity of aerobic respiration...
This simply discribes aerobic respiration process...

Anaerobic respiration
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It begins with Glycolysis process. The glucose molecule is broken down into two pyruvate molecules, releasing electrons in the process and generating two molecules of ATP..

Hope it helps you buddy