"This writer died long ago."
Translation:Этот писатель давно умер.
Well, first you used the feminine form of "погибнуть" and the masculine of "умереть". But besides that "умирать" is stylistically neutral, while "погибнуть" implies a tragic and sudden death, usually the results of accidents, attacks and wars. It's not applied to a death of old age, and is rarely applied to dying from illness (unless we are talking about a plague).