I attempt to lay out some of the key differences below. In the process I've had to paint a stereotyped and a bit caricatured picture of both medicine and psychology. I hope both parties will forgive me my license. I've perhaps used some confusing terms in my table. Let me explain in English.
Medicine takes a mechanistic approach to illness.
This means that the physician tends to view a patient in a detached way, as a body with a problem, much like an actual mechanic might try to figure out what was wrong with a car. The physician's goal is to understand the nature of the body's problem so that this problem can be addressed and fixed. The physician determines a diagnosis based upon her analysis of symptoms. Her prescription is designed to fix the diagnosed problem. Nowhere in this chain of events is the physician encouraged to be particularly interested in the experience of the person behind the problem.