The Roman Empire was not a "secular" polity in the modern sense (which is a fairly complicated notion anyway). Its political power rested on what we would define either "religion" or "ideology" and should be better understood, in modern terms, as being sort of both, like a lot of pre-modern systems of thought.
In this sense, the problem was not, strictly speaking, that being Christian was in itself illegal, which it wasn't, technically. The problem was that being Christian was inconsistent with what the Roman Empire perceived as an acceptable civic behavior and belief.
In other words, the Christian message, in its early form, was utterly unacceptable to the Roman ideology and, conversely, the Roman ideology was utterly unacceptable to early Christians.
Christianity and the Roman Empire can be seen as two systems to give meaning to one's life. At first, they were mutually exclusive. A bridge would be eventually found of course, and worked pretty nicely afterwards, but it required a hell of a lot of an effort (in time and ink and blood) to build it.
It is not just that Christians refused to pray to the Emperor, a refusal that was far from exclusively theirs anyway. It is more that Christians were not supposed to give a damn about the Emperor altogether, as the entire worldly existence was supposedly meaningless to them if not as preparation for the Hereafter. While to the traditional Roman worldview, worldly life was the only one that really made sense, though other concepts of the Hereafter existed (quite plenty of them in fact).
The Empire might have been more lenient in persecution (especially under, say, Decius) but the reasons why Christianity was considered superstitio prava immodica ("a malicious, indecent superstition") went pretty deep. Christians challenged the very basis of the traditional Roman moral and social, let alone religious, world.