Psychiatrist. A psychiatrist is a medical doctor while a psychologist is not. As a result they can more accurately review medical conditions that contribute or underly emotional illness. They can order labs, x-rays, specific neurological tests, and carefully consider the role of medications in symptom presentation. They can also perform physical examinations. Importantly, they can knowledgably prescibe meds.
Either. Either a competent psychologist or psychiatrist can effectively diagnose mental health disorders. It makes no difference. However when symptoms are severe enough it is wise to go to a psychiatrist for an evaluating first, especially if the symptoms are severe or life-threatening. Therapy is also strongly indicated as and optimal combination.
Diagnosis. A doctoral psychologist or equivalent trained in clinical psychopathology depending on the training program and exposure to large number of cases with mental illness should equally competent as a psychiatrist to make a diagnosis using the diagnostic statistic manual criteria. The key word here is exposure of a large number of patients suffering a mental illness.