How and why was the history of the byzantine empire different from that of the western roman empire?

Well, the Byzantine Empire was in a entire different time era, the Byzantine started around 500 all the way to 1400 while the west only lived around 150 years (395-476) I think. Also the byzantine spoke greek and had more of a greek culture while the west was filled with latin speaking and had a completely different culture. You should be comparing the Eastern Roman Empire to the west, which were around the same time era.

The answer is crop yield.