The correct comparison regarding the leadership of the Roman Catholic and the Eastern Orthodox Churches is: The Roman Catholic Church centralized authority in the pope, while the Eastern Orthodox Church shared power between the emperor and the patriarch.
In the Roman Catholic Church, the pope holds supreme authority over the entire church, while in the Eastern Orthodox Church, the power is more decentralized, with the patriarchs of the various national churches holding significant authority, often in conjunction with the emperor or other local rulers.