Way back after my graduation (in 1975), I got my first job as a clinical psychologist at the psychiatry department of a university hospital. There were three of us (psychologists) on the staff, plus numerous psychiatrists.
There often exists serious antagonism between psychologists and psychiatrists, but none of it existed in this place: It was a small town and we had truly good relationships.
One day a young psychiatrist told me: The difference between you and us is that you (psychologists) go to school for five years and study psychology, while we (psychiatrists) go to school for six years and study internal medicine, surgery, and all other disciplines of medicine, but only got two 1-credit semesters of psychiatry which, back then, we did not even take seriously.
In a more serious way, there is a big difference between psychologists and psychiatrists because there is a big difference between psychology and psychiatry.
Psychology has its roots in philosophy. For millennia philosophers have attempted to study the human mind, our motivation, our emotions, etc.
In 1879, Wilhelm Wundt did the first science-style experiment which studied the correlation between the weight of an object placed into a human's hand and the feeling produced by the object. Wundt was able to show mathematically that if the object doubles its weight, the feeling doubles. He also determined that this was true only for small objects, once the object became too heavy, the subjects were not able to tell that one object was twice the weight of another just by feeling them with their hands.
With that, psychology was born in 1879 Germany. It was no longer a branch of philosophy but an experimental discipline. It was very important to the early psychologists to be accepted as scientists as opposed to "mere" philosophers.
Psychology is the study of the mind. Mostly the human mind, but many psychologists specialize in the study of animal mind as well. In America, especially in the early 20th Century, psychology often is/was seen more as the study of human behavior without making any conclusions about the mind. This, in my opinion, has greatly reduced the usefulness of psychology in America, at least in its (psychology's) early history. However, that has changed, and many American psychologists do study the mind rather than just external behavior.
Psychology then studies such things as intelligence, motivation, emotions, learning, and other aspects of the mind. Psychological research is typically done in lab conditions, studying two groups of subjects (a test group and a control group), which are typically presented with a specific task under specific circumstances. The researcher then measures the subjects' reaction, or performance, or whatever else it is he is studying. He then uses statistical analysis to find correlations between some kind of stimulus or condition and some kind of reaction or result.
Psychiatry is quite different. It is a branch of medicine. Its goal is to study and cure mental diseases. It is not interested in the mind the way psychology is. Rather, it is interested in finding the underlying physiological problem: A chemical imbalance in the brain of the schizophrenic, and things like that. Once the physical cause is found, and equally physical solution is looked for, e.g., Prozac.
There is some natural overlap between psychology and psychiatry: Psychology as the study of the mind includes the study of disturbed mind, even sick mind (psychopathology). Both, psychologists and psychiatrists study psychopathology.
That means, among other things, that both psychologists and psychiatrists may be dealing with people who have problems. They will be dealing with them in completely different way. The psychologist will work with the mind. He will use psychological means: Counseling, hypnosis, psychotherapy, relaxation, etc.
The psychiatrist will treat the body: He will most likely prescribe drugs. There are psychiatrists who use other methods. But they still are medical doctors at heart.
What that means when you are seeking help for someone can be simplified into something like this: If his problems are physiological (schizophrenia and other psychosis), take him to a psychiatrist. If his problems are mental or behavioral (hyperactive kid, marriage problems, etc), take him to a psychologist.