The difference between these two theories has to do with their ideas about how the diseases are contracted -- how people come to be sick with the disease.
The miasma theory of disease held that diseases are caused by "bad" air. If people are exposed to bad air, they breathe it in and become sick. In this theory, diseases happen because of the air and not for any other reason.
In the contagion theory, the diseases are actually passed from person to person. This does not really tell us anything (not necessarily, at least) about the ultimate cause of the disease, but it does tell us that once Person A has the disease, he can spread it to Person B and C and beyond. B and C will not have to go and breathe the "bad" air.