Short answer is that psychiatrists are medical doctors and (more often than not) medical researchers, and thus psychiatrists themselves often do not practice psychotherapy.
Psychiatrists are physicians, which means they have received a medical degree (M.D.) and have completed medical training, along with an additional four-year residency in psychiatry. Some psychiatrists pursue additional fellowship training after their residency. Thus, it takes at least twelve years of practice to become a psychiatrist.
Psychiatrists are legally allowed to prescribe psychotropic medications to patients. This is because they are medically trained to do so, having completed a medical degree and medical training. A large portion of this medical training is dedicated to learning how psychotropic medications work in the brain and in the body, particularly in those of patients with mental illnesses. For this reason, psychiatrists are generally more aware of the physical symptoms of illness (much like a doctor would be) as well as the mental symptoms, and thus must make an informed evaluation to determine whether the patients' symptoms are due to physical or mental factors (or both).
Though psychiatrists can perform psychotherapy, the vast majority are purely medical researchers. A psychiatrist may thus refer a patient to a clinical psychologist or a specifically-trained therapist for their individual psychotherapy needs. A physician or psychiatrist (both trained in psychopharmacology by definition) who both practices psychotherapy and prescribes psychoactive medication may refer to themselves as a "psychopharmacologist", though this is more of a distinction than a formal occupation.
Clinical psychologists are PhDs who have received several years of graduate training in the science and theory of psychology-based dysfunction and mental illness. They do not receive medical training, but they do undergo 2-4 years of supervised clinical practice before they are licensed as clinical psychologists.
A clinical psychologist's job description is a bit more broad than that of a psychiatrist. While many provide psychotherapy to patients suffering from mental illness, other clinical psychologists may opt to teach, administer psychological testing, conduct research, work with administrators, develop treatment and prevention programs, or work as psychological consultants.
Though there is very little that a clinical psychologist can do that a psychiatrist simply can't do, a clinical psychologist is trained in mental health and well-being first and foremost, and thus is typically called on to perform tasks related to mental health (such as psychotherapy or psychometric testing). The difference may be comparable to that of a professor versus a teacher. Though the professor typically undergoes more training, they split their focus between their research and their students (and may prioritize research first), whereas high school teachers are often teachers through-and-through, and thus their primary focus is on the students. Thus the teacher role is a bit more interpersonal.