So, in short, the Byzantines continued the Roman legacy of empire and war and law for almost a thousand years after Romulus Augustus was driven out of Rome. The Byzantines may not have spoken Latin, and few of their emperors came from Rome, but in most important ways, they were Romans. Except one really important way. The Byzantines followed a different form of Christianity, the branch we now call Eastern, or sometimes Greek, Orthodox. How there came to be a split between the Catholic and Orthodox traditions is complicated. You might even call it Byzantine. What matters for us are the differences between the churches, the main doctrinal one being about the dating of Easter, and the main political one being about who rules whom.