Nicked from wikipedia, but covers the differences pretty much:
Syndromes, illness and disease
Medical usage sometimes distinguishes a disease, which has a known specific cause or causes (called its etiology), from a syndrome, which is a collection of signs or symptoms that occur together. However, many conditions have been identified, yet continue to be referred to as "syndromes". Furthermore, numerous conditions of unknown etiology are referred to as "diseases" in many contexts.
Illness, although often used to mean disease, can also refer to a person's perception of their health, regardless of whether they in fact have a disease. A person without any disease may feel unhealthy and believe he has an illness. Another person may feel healthy and believe he does not have an illness even though he may have a disease such as dangerously high blood pressure which may lead to a fatal heart attack or stroke.
Disease can be thought of as the presence of pathology, which can occur with or without subjective feelings of being unwell or social recognition of that state; Illness as the subjective state of "unwellness", which can occur independently of, or in conjunction with disease or sickness; and sickness as the social classification of someone deemed diseased, which can also occur independently of the presence or absence of disease or illness. So someone with undetected high blood pressure who feels well would be diseased, but not ill or sick. Someone with a diagnosis of late-stage cancer would be diseased, probably feeling quite ill, and recognized by others as sick. A person incarcerated in a totalitarian psychiatric hospital for political purposes could arguably be then said to not be diseased, nor ill, but only classified as sick by the rulers of a society with which the person did not agree. Having had a bad day after a night of excess, one might feel ill, but you would not be diseased, nor is it likely you could convince your boss to recognize you as sick.
Sickness confers the social legitimization of certain benefits, such as illness benefit, work avoidance and being looked after by others. In return, there is an obligation on the sick person to seek treatment and work to become well once more.
So with that in mind I'd say that alcoholism is a disease.