A screening test is given to those who have no symptoms of the condition in question. An example would be measurement of cholesterol levels in people who have no symptoms of cardiovascular disease. When a person does have symptoms, however, the picture changes, as the chance of uncovering something meaningful to health is considerably more likely once a person has a symptom or is at higher risk of a condition. Risk is best determined by a doctor when he or she checks your physical condition and "takes a history" -- asks about your signs, symptoms, and medical history in a complete and thorough way.
A diagnostic test is used to confirm a suspected condition once initial testing has revealed its possibility. Diagnostic tests are often more expensive and risky than initial tests. However, their usefulness should far outweigh their risk, or they should not be undertaken. An example of a diagnostic test would be cardiac catheterization (threading of a tube into the heart) if serious heart disease is suspected.
