It's not really a good comparison to make between Roman Empire persecution of Christians and persecution by Byzantine Orthodoxy. The reason it's not a good comparison is that not only are the social situations entirely different, but the religious outlook of both settings is entirely different.

In the Roman context religion is much more performative and societal, than credal and individual. So one of the major stumbling blocks for Graeco-Roman polytheists is that Christians don't simply worship another god, they refuse to worship the rest of the gods, refuse to give divine honours to the emperors, refuse to participate in community religious practices, and so rupture the social order, risk the wrath of the gods, and subvert the religio-political order.

In the Byzantine context you are talking about, now the ruling order is steeped in a Christian theological tradition that sees things differently, and so right practice is not enough, right belief is crucial. By the time you get to the debates about Christ's nature in the 5th century, the church has just been through a long, protracted, and heated debate about the relationship between Christ and God the Father, and so it is well-honed in why right-belief matters, even if the liturgical and worship practices of two groups are very similar.

You have also seen the development of the importance of church councils as authoritative means of legislating doctrine into existence. The way the late fourth century treats Nicaea changes the playing field, in that creeds were not really used as 'weapons of orthodoxy' before the late fourth century. But not theologians will appeal to the authority of creeds and councils, and call for a council as a way to resolve debates.

The mid-fifth century council of Chalcedon (451) represented the kind of arrival-at-a-doctrinal-definition that 381 did for the earlier issue of relating the Father and the Son in Christian thought, except Chalcedon is dealing with how to relate Christ's metaphysical human nature and his metaphysical divine nature. Groups that refused to adhere to this decision are groups that were on either extreme, either 'Nestorian', holding some form of duality that treated Christ as two entitites, or 'Monophysite', holding that the two natures were fused into some form of a new, third, nature. Both those definitions are gross simplifications, and part of the issue of discussing orthodoxy and heresy in the 5th century is discussing fine-grain details of theological metaphysics, dealing with technical language that is not necessarily uniform across all the players, and is developing over time, and also wilful misrepresentation or misinterpretation of opponents' beliefs. That said, Monophysitism was centered in Egypt, Nestorianism came to be centered in the Eastern church (by which I mean Antioch and east, into Syria; it became the dominant form of Christianity to spread outside the Empire into Central and East Asia). I don't really like the term Nestorianism because it's probably not accurate of Nestorius' views, and given the emerging linguistic difficulties in Greek-Syriac theological relations it may not represent the Syriac churches' views accurately either.

Back to the question at hand, yes, Byzantine persecution/suppression really was founded first upon theological convictions, this was a period in which wrong belief put you outside the church and was seen to lead to eternal damnation, so it was no minor matter. Growing cultural and linguistic divisions with these groups may have contributed to antagonistic relations, but the first and prime reason is convictions about how Christ's natures related to each other.