They are similar but sickness is more general than disease. You could say, "His prolonged sickness was caused by Alzheimer's disease." He was sick, meaning he had symptoms of an illness. The illness was the result of the disease. Hopefully that clears it up. Influenza, also known as the flu, is a contagious disease that is caused by the influenza virus.

The flu can be classified as a sickness but cancer of the lungs is a disease.