One of the principal grievances that the Roman authorities had against the early Christians was their refusal to swear allegiance to the Roman emperor and state deities such as Jupiter. Roman religion embraced a multiplicity of gods and goddesses, and, far from rejecting foreign deities, the Romans had actually adopted them when this seemed desirable. Such an elastic attitude made it hard for the Romans of the first centuries of our era to understand the mentality of Christians with their claims to exclusive possession of religious truth. No Roman divinity was endowed with the mind-reading ability of the Christian [God], and consequently, it did not matter for Romans what private beliefs an individual might hold. Christians could not accept this. Their obstinacy in rejecting any religious compromise that might indicate acceptance of other gods made them appear as a group of dissidents who posed a threat to the state by not accepting its authority. This, rather than religious intolerance, was the main reason for their persecution. (Complete New Testament Greek, "Historical Insight," p. 133; emphasis mine).
In short, according to Betts, early Christians were persecuted for being "a group of dissidents who posed a threat to the state by not accepting its authority" (as if this were somewhat more understandable than religious persecution!). Indeed, Betts appears to be saying it was the Christians who were guilty of religious intolerance because of "their claims to exclusive possession of religious truth."
However, New Testament scholar Larry W. Hurtado seems to see it differently. According to Hurtado, the early Christians faced opposition precisely because of their religious beliefs. In fact, somewhat ironically, Christians were often accused of being atheists! As Hurtado tells us:
Early Christians were atheists! At least, that's how some people of the time viewed them in the earliest centuries, and it's not difficult to see why. Most importantly, they refused to worship the traditional gods. But also, judged by Roman-era criteria, they didn't even seem to practice a recognizable form of religion. In the crucial first couple of centuries at least, they had no shrines or temples, no altars or images, and no sacrificial rites or priesthood...
Unlike the emphasis today, however, in the Roman world atheism wasn't primarily a matter of belief or unbelief. Instead, what counted then as "piety" or being religious was mainly participation in worshiping the gods. In that setting, to refuse to do so was atheism...
To appreciate what this rejection of the traditional gods meant, we also have to understand that gods and reverencing them were woven through every aspect of life. Families had household deities. Cities had their guardian gods. The Roman empire at large rested upon the gods, such as the goddess Roma...
So, to refuse in joining in worshiping any of these deities in a thorough-going manner was a very radical move, and a risky one too, with wide-ranging social costs. People understandably took offense, and Christians could be in for a great deal of anger and hostility that might include verbal and physical abuse. In some cases, the Christian rejection of the gods led to arraignment before Roman magistrates, resulting in punishments, even executions. By the third century, there were occasional spasms of imperial persecution against Christians that could include confiscations of possessions and death sentences... ("When Christians Were Atheists" at larryhurtado.wordpress.com)
As Hurtado tells it, the early Christians were persecuted for their beliefs. Moreover, it was the non-Christians, including the Roman government, that were religiously intolerant.
How to resolve these two seemingly contradictory viewpoints? Perhaps the easiest way to resolve them is to say that Christian believers in the Roman empire were viewed as both politically and religiously subversive. Moreover, since, as Hurtado points out, the "Roman empire at large rested upon the gods," we can say that religion and the state were unavoidably intertwined. Thus, to reject the existing religious system was, in effect, also to reject the existing political order. Or, at least, that is how the Roman authorities saw it. True, the Roman government tolerated the existence of Judaism, and Judaism shared Christianity's strict monotheism. However, Judaism could be tolerated because it was associated with a specific ethnic identity. In contrast, as Hurtado points out, "Christianity quickly emerged as a trans-ethnic movement, aggressively proclaiming its message and recruiting former pagans to its peculiar message on a scale that made it a threat in a way that was never true of Judaism" (emphasis mine). As a result, the persecution of Christians was possibly inevitable. We can see this same dynamic today in countries like China, where the government engages in efforts to control the spread of Christianity, in part because Christian beliefs pose a challenge to Communist ideology (hence the effort to "sinocize" Christian doctrines to make them more consistent with that ideology), but also in part because Christian churches are seen as posing a threat to the absolute power of the state.
Statue of the emperor Nero, early persecutor of Christians. Image from Wikimedia Commons.