Asthma and emphysema are two entirely different diseases with some symptoms in common. Asthma is a spastic and inflammatory disease of the airways that causes reversible obstruction of the bronchial tubes. It usually responds to treatment.
Emphysema is a disease of the lung tissue, especially the alveoli (air sacs) at the end of the bronchial tubes. Emphysema results from destruction of these alveoli. This process is not reversible and responds less well to medical treatment. Emphysema is slowly progressive; you will get short of breath and become more disabled as time goes on. Eventually you may require continuous concentrated oxygen to be comfortable. This irreversible disease (and other obstructive lung diseases such as chronic bronchitis) is often referred to as chronic obstructive pulmonary disease (COPD).
The majority of COPD individuals have a long history of smoking cigarettes. It is possible to suffer from both COPD and asthma. Rigorous treatment of the asthmatic component can often significantly improve your quality of life.
In recent years, some asthmatic individuals have developed an irreversible obstruction of their bronchi and appear to have a syndrome that is very much like the fixed obstructive disease seen in COPD.