Why did the Romans worry about the spread of Christianity?

Christianity seems to have been both ignored and tolerated until the third century, when there began to be concerns about the loyalty of Christians and the risk that Christianity posed to the good government of the empire.

The real concern was that the Christian Church seemed to have become a powerful state within a state. It passed and enforced its own laws against Christians, kept its own treasury, and Christians appeared to owe allegiance to the bishops ahead of their Roman governors.

There were troubling examples of treasonous behaviour by some of the Christians, that served to alienate the emperors and to justify some form of persecution before the Christians got out of hand. On the day of a public festival, Marcellus the centurion threw away his arms, and the ensigns of his office, and exclaimed with a loud voice that he would obey none but Jesus Christ the eternal King, and that he renounced forever the use of carnal weapons and the service of an idolatrous master. The soldiers, as soon as they recovered from their astonishment, secured the person of Marcellus. By his own confession, he was condemned and beheaded for the crime of desertion. Examples of such a nature are more akin to martial or even civil law than religious persecution.

In 303 CE, Galerius and Diocletian decided, for what they saw as the good of the empire, to institute the Great Persecution of Christianity. Diocletian abdicated in 305 CE, after which persecution ended in the West, but it continued until 311 in the eastern part of the Roman Empire.