This is a bit of a controversial question, and I have come across different beliefs about the Great Schism. For example, this comment thread. Some Catholic and other non-Orthodox Christians say that the Eastern Orthodox Church "veered away" from the early Church. Some Eastern Orthodox and other non-Catholic Christians say that the Roman Catholic Church "veered away" from the early Church.
However, it would be nice if someone were to set the record straight about it.
Did the RCC or EOC "veered away" from the early church? Which church became very different from the early Church theologically, practically, culturally, or structurally? Or is it something more complicated?
Both, neither, it's more complicated (of course!)
If you ask a RCC apologist, they'll tell you that the RCC is the one true church and the EOC veered off; and vice versa.
But really, the early Christian movement from its very inception was a group that was subject to historical change and development. There's no, "frozen in time moment" of pristine "Early Christianity". There's snapshots of the nascent Jesus movement as it begins in the aftermath of Jesus' death (and resurrection, as they obviously believed), and the New Testament is a series of "slices" of what that movement looked like between roughly the 50s and 90s AD. And even within those documents there is change, most notably Gentile inclusion and the theological elements that brings.
As you move outside the New Testament period, it's clear that things keep developing, organically and subject to historical forces (as with all movements!). You do see a considerable unity and uniformity across church centres in different cities, in both Latin West and Greek East (and to some extent beyond the border of the Roman Empire and further East, in Syriac and other Eastern Christianities). And both RCC and EOCs have a theological framework of development built into them - they are not atavistic "here is pristine early christianity and we are it".
The Schism of 1054 comes as the result of a long process of drift by both churches away from each other, but they don't conceive of themselves as different churches. They are one church. However, the ongoing linguistic and cultural divide tends to exacerbate that drift, because when you aren't talking to half the empire, and don't know what's going on there, you naturally are not quite on the same track.
The RCC of course claims papal supremacy, and this is a doctrine that only really emerges in the 5th century and onwards, and develops in strength. The claim of Rome to sole power is not one that the eastern churches took very kindly to, and one the continually rejected. The emergence of a hierarchy at the urban level, bishop-priests-deacons, seems to have occurred very early, by the 2nd century, but the development of hierarchy among the bishops of different cities takes much longer to come to fruition.
In essence, there's no answer to your question because it can't be formulated in those terms and continue to be sensible. By 1054 they have diverged from each other but they are one church which has a historical trajectory that leads back to the early church, and claims to be more pristine or to have 'not veered off' are largely self-serving, not historical.