What's the difference between neurological disorders in the brain and mental illnesses?

Psychiatry is differentiated from neurology by a totally arbitrary line, and there is a large degree of overlap. Generally this line is drawn at the point where a physical manifestation of the disease is not obvious(psychiatry), versus where there is a somatic component. Neurology deals with more organic brain diseases, with more obvious brain pathologies, and generally larger involvement of sensory and motor systems. Psychiatry on the other hand deals more with constructs related to emotion, executive function, and beliefs that have less distinguishable and obvious pathologies.

However, it has not always been this way, and historically neurology and psychiatry have been a unified field. However in the 20th century, as schism occured. One interesting consequence of this is the use of different terms to describe the same phenomenon, such as aboulia vs avolition

Despite this, it is becoming more and more clear that neurology and psychiatry are interrelated. For example, mania, depression, and other “psychiatric” phenomenon are very common in strokes, largely treated by neurologist. Manifestation of “psychiatric” symptoms are not uncommon in neurological disorders, with depression, and anxiety being extremely common, although not always necessarily pathological, or abnormal. As psychiatry becomes more evidenced based and a focus on biomarkers of disease develop, neurology and psychiatry are being integrated.

Reference: Alexander Saytsev