I can answer this for between c.500 and 700, though I would argue that these differences did not necessarily mean that there were entrenched divisions between different strands of Christianity already. Nor were they inevitable during the Great Schism for that matter, since as far as I know there is a tendency for historians to de-emphasise its importance in recent studies - I'm not an expert on this however so you will have to wait for another person to talk about that I'm afraid!

First things first, I think it would be incorrect to describe the churches of late antiquity as 'Catholic' or 'Orthodox', as these labels would not have been understood by people at the time. Instead, I think people in both Rome and Constantinople would have understood their beliefs as part of 'Niceno-Chalcedonian' Christianity, due to the decisive impact of the Councils of Nicaea and Chalcedon. These were the beliefs of the clergy in these two cities, but the picture gets way more complicated when we look at the wider context. Rome was then part of Ostrogothic Italy and ruled by Theodoric the Great, an 'Arian', whilst Constantinople was ruled by Anastasius I, a sympathiser of miaphysitism. To make a long story short, the Council of Chalcedon in 451 had left many people in Syria and Egypt, now known as miaphysites, disappointed. Emperor Zeno tried to accommodate their beliefs and endeavoured to limit the impact of Chalcedon, but this made the papacy very angry, since Pope Leo made a significant contribution to the council, which led to the Acacian Schism that lasted from 484 to 518, as the pope anathematised the patriarch of Constantinople, Acacius, for the latter's acceptance of Zeno's compromises. Zeno's successor was, as I said, a closet miaphysite, so this schism would have to wait until the ascension of the Chalcedonian Justin I in 518 to be healed.

There were therefore plenty of differences, but I don't think we can describe them as irreconcilable. Indeed, the fact that many Niceno-Chalcedonian churchmen and secular leaders were content to be ruled by 'heretics' would surely suggest that religious beliefs were instead more flexible and that drawing such strict boundaries obscures their faith's dynamic nature. 'Arians' continued to serve in the emperor's armies, even when Constantinople was fighting the 'Arian' Vandals and Ostrogoths, whilst popes and western bishops had likewise loyally served their 'Arian' 'barbarian' masters. Refugees from 'Arian' persecution, such as those from North Africa, were also welcomed in Constantinople during the Acacian Schism, which further supports my case. As for the dispute itself, whilst the papacy refused to back down over its vehement support for Chalcedon, it is difficult to find evidence that western churchmen elsewhere had equally strong feelings against their eastern counterparts. This schism was certainly an important event, but I'm wary about it marking a significant moment in the history of east-west relations.

As you know, in the 530s Justinian attempted to reconquer the west and two decades later had finally achieved a semblance of control over Italy, North Africa, and parts of Spain. At this point, Chalcedonianism had been the centrepiece of imperial doctrine for a long time, but other beliefs were still present - the empresses Theodora and Sophia (wife of Justinian's successor, Justin II) were for example miaphysites, whilst Justinian himself continued to try to find a compromise that would unite the miaphysite and Chalcedonian churches. The picture in the west was similarly complicated. In Italy 'Arianism' was still not extinguished, and in Spain 'Arian' kings continued to call the shots. By 553 Justinian hit upon a solution to his miaphysite headache - by condemning some teachings used at Chalcedon to support the 'orthodox' cause, he hoped to bring back moderate miaphysites back into the imperial fold. Unfortunately, this made the west very angry. Part of this was due to the campaigns of a few energetic bishops and the prominent role played by the pope (Pope Vigilius initially opposed it, but was eventually bullied into agreeing with Justinian), but I think the crisis also escalated because the empire was now on the west's doorsteps and so its theological disputes were no longer distant stories, but tangible debates that had to be engaged with. For these reasons, the west (or at least, the non-'Arian' part) stuck by its guns over Chalcedon and although much of Roman Italy eventually agreed with imperial policy, a few angry bishoprics in northern Italy refused to be in communion with the papacy until the eighth century over this.

As before with the Acacian Schism, this dispute, the 'Three Chapters' Controversy, was a complicated affair as well in terms of allegiances, so again we can't generalise it into a solid difference between Roman and Constantinopolitan doctrines. In any case, the papacy eventually acquiesced to imperial pressure, so officially the city of Rome followed the Constantinopolitan line. Not that it meant much anyway, since most miaphysites were not impressed by this half-hearted compromise and continued to develop a parallel church hierarchy in the empire's eastern provinces. Justinian himself in the last year of his life became a believer in the 'Julianist' brand of miaphysitism; so much for being a paragon of 'orthodoxy'!

I would argue that much of the same continued into the early seventh century. By then, the pope had essentially become an imperial official who helped with the difficult task of governing Italy, so there is little evidence of any pope bringing up major issues (though there were minor ones, such as Pope Gregory the Great being angry at the patriarch of Constantinople for styling himself as 'ecumenical'). It is also worth pointing out here that the empire did not lose control of Italy at this point. Half of it was indeed lost after the Lombard invasion soon after Justinian's death, but key regions around Ravenna, Rome, and Sicily were still held by imperial forces; indeed, they would remain Roman possessions until the eighth century. Imperial forces now confronted the 'Arian' Lombards (though they were not uniformly 'Arian') across the peninsula, so the papacy probably had little interest in stirring up theological trouble during this time of crisis.

This however changed in the 630s. In short, Emperor Heraclius embarked on a programme of religious unification, one that succeeded in getting some miaphysites (and even some 'Nestorians') to rejoin the imperial church by promoting the doctrine of monenergism and later monotheletism. The latter actually received support from Pope Honorius, but shortly afterwards the papacy turned against this and along with an influential group of Palestinian monks, began to campaign for its repudiation. This resistance drew in both bishops from beyond the empire as well as secular leaders within it, most importantly the exarchs of North Africa and Italy, so it seems to be a pretty good example of the west opposing the east. However, as I have mentioned already, the anti-monothelete movement also had its eastern backers and arguably the co-emperor Theodosius was involved too, so it can hardly be reduced to a simple dichotomy. In any case, by 655 Constans II's agents had arrested Pope Martin and Maximus the Confessor, the principal Greek agitator, and sent them into exile. Though anti-monothelete sentiments remained, there would no longer be any obvious signs of rebellion over this issue.

Our understanding of the next few decades is very unclear, but it seems that the papacy still harboured anti-monothelete elements and that eventually similar groups in Constantinople came to power as well, which led Constantine IV to repudiate monotheletism and thus officially return to 'orthodoxy' in 680/1. However, troubles would continue over the next few decades, since Justinian II's synod over clerical behaviour was not accepted in Rome and as a result the pope was nearly arrested. Even later in the eighth century Leo III probably tried to arrest the pope as well, perhaps due to the emperor's iconoclastic leanings, though modern scholars are now emphasising the political and financial aspects of this whole thing rather than just the theological dispute (if iconoclasm was even a thing under Leo).

As I have hopefully illustrated, the western and eastern churches were not necessarily divided, but nor were they united. There were many issues in play and I have only touched on the tip of the iceberg, with many other internal differences all playing a role in the colourful tapestry of Christendom at this point. Indeed, as Peter Brown had put it, Europe in late antiquity was a world of 'micro-Christendoms', each distinctive from each other and pursued different goals. I think this concept can be clarified a lot more, but his emphasis on the diversity of beliefs in this period is surely correct, so it is difficult, if not impossible, to see the complicated disputes of late antiquity as signalling later divisions that occurred in very different contexts. There were certainly similarities, but I suspect that it would be difficult to draw any direct and unbroken links between these issues over the centuries.

If I have to make a general conclusion, I would argue that both the papacy and the church in Constantinople almost always operated in reference to the imperial agenda; they often reacted in different ways to different emperors' policies, but their actions were still molded by the emperor's actions. For them, the empire never fell, so it was natural for them to still view events through an imperial lens, even if they had to resort to anathemas or outright revolts to get their points across.