What aspects of ancient Roman military tactics and style did the Byzantine Empire keep over the centuries?
The military of the two empires changed a great deal over the centuries even within each empire. There were more differences than similarities. The Romans were always a predominantly infantry army, while the Byzantines were famous for their heavy cavalry. Two similarities that I can think of are that both relied on highly disciplined and trained troops, as opposed to massive numbers or personal bravery or fanaticism. Also, they both liked to use the most modern available siege equipment, including those lighter types of balista’s and catapults used in field battles.