There are two parts to your question which probably need separating: the description and recognition of symptoms, and treatment and aetiology. Ancient Greek writers became really quite good at describing symptoms of disease in the early Classical period, when the Hippocratic authors were collected together and more broadly the genres of history and ethnography emerged. Thucydides' description of a plague to hit Athens about 430 BC, in book two of his history, is incredibly detailed, to the point where modern epidemiologists can debate the likely strains of disease that it might have been.

With respect to treatment, they didn't assume all illnesses were the same, but there was little explicit methodology on treatment. The Hippocratic corpus collects writing on a number of illnesses that still occur in a similar way today, like epilepsy and (controversial) depression. When you reach Galen, who was writing in the second century A.D., there were more common methods, but that's outside the scope of your question I suppose.