How did the political and religious ideas of the byzantine empire compare with the western empire?

In the restored western empire the secualr and religious authority was separated according to augustinian philosphy in the byzantium the dominant politicla philosphy was cesaropapism where emperoro was also a religious leader

They made a checks and balance system to equal power within the branches. the "check" meant that one branch would watch one branch and "balance" them if they were too powerful. if you want more info on this, i would just google checks and balance system.