While both professions deal with hair, their jobs are quite different. A trichologist studies hair structure, function, and disease. A trichologist studies some parts of anatomy and physiology as it relates to the hair because a lot of what happens with our hair begins in the rest of our bodies.
They can tell by the examining the hair and scalp the root causes of most conditions. More importantly, it is a specialized branch of medicine dealing with hair health.
Hair stylists, alternatively, do not know the physiological causes of the above conditions. A hair stylist, typically, only handles the cosmetic part of dealing with hair, even though they might study some parts of human biology.
Stylist understand the best way to shape, trim, and cut hair to suit the face, and while the stylist might know some parts of hair structure, they usually only understand topical issues, for example fixing split ends.
Ultimately, people with hair loss, problems with the scalp, hair texture problems should consult a trichologist. Someone with bad dye job or a bad haircut should consult a hair stylist.