The big schism, the division between the Catholic west and Orthodox east, would not happen. That evolves out of head-butting politics between the Papal sphere of influence and the Imperial sphere, but in this world the two spheres would be more or less one and the same. You would probably get uppity Popes from time to time trying to insist their word comes before the other patriarchs, but with Imperial power present in the west it is unlikely any Germanic kingdoms would accept Papal primacy since it would do them no good. Remember, accepting Papal primacy was how Charlemagne got made an Emperor.

That means that the worlds "Catholic" and "Orthodox" will be interchangeable words for the Chalcedonian dogma. 

However, as Aaron said, there was a Monophysite split between Alexandria and the Orthodoxy that would have happened regardless, and they became very popular on the eastern fringes. You'd see it entrenched in Egypt and East-Africa, popular in Armenia, and influential in Syria. Also in Syria, particularly deeper in Christian parts of the otherwise Zoroastrian Persian Empire, you would have the Nestorians. Historically many of the Germans were Arian as well, though whether or not Arianism would have lasted in this situation is hard to tell. Miaphysitism happens later, and represents the moderation of the Monophysites. 

Now, the meaning of these terms seems really petty to me and probably most modern people, but it was important to people in this time period. I will try to explain them though.

The Orthodox position is that Jesus was both divine and human at the same time, and that this divinity and humanity were distinct but acting in tandem.

The Monophysites held that Jesus was divine and that was that. One nature, not two.

The Miaphysites who later evolve from the Monophysites and represented by the modern Coptic churches held that Jesus had one nature, but that this one nature was some sort of mix between divinity and humanity. 

The Nestorians, who are the Syriac christians of the modern day, believed that Christ was bother divine and human but that these two aspects were completely separate. They are different from the Orthodox because the Orthodoxy saw the humanity and divinity as being equally present, the Nestorians saw Jesus as a human with divinity inserted post-production.

The Arians believed that Jesus was the son of god, but that he was not divine, but was a separate subordinate human son kind of like a Greek demi-god.

There is also the Donatists, who you will see mentioned if you research this stuff. They probably would not be present in the RP, as they were mostly worried about the nature of apostasy, which wouldn't be common in the 7th century. They were popular when large parts of the western world were still hostile to Christianity, and they represented the faction that believed that anybody who renounced Jesus, even under duress, could never be forgiven or allowed to receive the Eucharist. It is possible this idea might resurge in the face of Islam.

Now, if there is anybody better at theology than me, please speak up, because the theological divides make very little sense to me and I don't understand why anybody could ever have cared. This was shit that nearly brought the Empire to civil war several times, and it would have divided everything from rival street gangs to aristocratic families. And that I truly don't get.