A neurologist is a physician who diagnoses and treats disorders of the nervous system which is comprised of the brain, spinal cord and nerves. These doctors do not perform surgery, but refer patients to neurological surgeons when they determine that surgical intervention is necessary.
Some of the conditions that neurologists diagnose and treat are epilepsy, aneurysms, hydrocephalus, Parkinson's disease, multiple sclerosis, stroke, spinal disc herniation, and spinal disease.
In addition to using diagnostic tests like MRI, CT scans, EEG and EMG, neurologists also employ neurological testing to gauge muscle strength and movement, balance, reflexes, sensation, memory, speech, and other cognitive abilities.

A psychiatrist is a doctor with specific training in the diagnosis and treatment of mental illness.
He or she can not only provide the counseling necessary to both diagnose and treat a patient, but can also prescribe medication when needed. In some cases, a psychiatrist will only provide the medication and the counseling will be provided by another healthcare specialist, like a certified counselor or psychologist.
Like other doctors, psychiatrists employ diagnostic tools like CT scans and MRI in order to observe the structure and function of a patient's brain.
Once a diagnosis is made, these specialists may use behavior or cognitive therapy in order to address the patient's condition, or a multitude of other types of therapy, in conjunction with or in place of medication.