The Catholic Church holds onto both papal supremacy and papal primacy, while the Eastern Orthodox Church and the Oriental Orthodox Church hold onto papal primacy only, and with a different idea of how this should be applied. It is difficult for both Catholics and Orthodox to agree because those are two traditions that grew separately (it's not as if Rome suddenly claimed to be the head of the Church in 1054), but what can be said from a purely historical/secular perspective?
Hey! Not to discourage further discussion and responses, but I answered a similar question a little ways back. In it I give a brief history of the papacy from the 5th through the 11th centuries with a focus on its changing role within Latin Christianity. Happy to answer any follow-ups you may have :)