Christianity preached exclusive monotheism, forbidding its members from participating in the rituals of the state religious cult. These rituals were understood as being vital to the maintenance and continuation of the Roman State. By not participating in them, Christians were, in effect, actively undermining the Empire itself. Thus, to be a Christian was in a sense to be a traitor.

Jews were also criticized for this, but Judaism had a respectably ancient pedigree (prior to the 17th century or so, novelty was considered a very bad thing and being rooted in ancient tradition a very good thing, many early Christian apologetics appeal to the antiquity of Jewish prophecies to shore up against accusations of novelty). Christians, on the other hand, were both new and "odd."

As the Empire teetered during the Crisis of the Third Century and as it emerged from that crisis during Diocletian's reign, Emperors often turned to religious rituals to promote unity through the empire. Sometimes persecutions weren't so much explicitly directed at Christians, as directed towards those who refused to take part in these rituals.

The Christians generally denounced all Roman gods as being demons. Unlike other religions they were constantly trying to recruit people.

Also, philosophically speaking, Christianity was unlike any other pagan religion. Having a single god that can not even be comprehended (versus a bunch of gods that were basically representations of human characteristics) was quite a peculiar notion a the time.

Edit: For more on the topic, read The Origin of Satan by Elaine Pagels.

The early relationship between Christianity and the Roman Empire was turbulent, to say the least. When we examine Roman culture (religious and otherwise) and the aspects of Christianity at the time (~100-300CE), lots of good reasons for this become clear.

As others have stated, the Empire was generally accepting of a variety of religions as long as they agreed to "play nice" with the diverse beliefs present in their society. Even Judaism, with all its persecutions by Roman authorities and awkward insistence that all other religions were untrue, was relatively grudgingly tolerated in part due to its ancient pedigree. It's strange to think of Rome as truly tolerating Judaism, but its inclusion of the faith and its people had long been contradictory: it's probable the Empire destroyed the Temple of Jerusalem, center of Jewish society and worship, yet adopted the Jewish division of the week into seven days (instead of the traditional Roman eight) in the same century. Since the Romans saw themselves as the inheritors of Greek culture and sought to emulate them (and the other ancient cultures they absorbed, to lesser extents), it's unsurprising the government would have a certain respect for Judaism's long history. Christianity, a cult in its infancy, accorded no such respect, regardless of its claims as the successor of Judaism.

Early Roman Christianity also rejected the social norms of many other important aspects of Roman society. "Christians cut across the normal courtesies of observing the imperial cult and that made them a potential force for disruption in Roman life. Indeed, the language they used in their enthusiasm for their saviour seems almost to be borrowed from the language which the imperial cult was developing in the lifetime of Jesus. So a Greek inscription found in Ephesus calls Julius Caesar 'god made manifest'; the Emperor Augustus's birthday was 'good news' and his arrival in a city the 'parousia' - exactly the same word which Christians used for Christ's expected return. It would be easy for sensitive Romans to hear such Christian usages as deliberate and aggressive plagiarism" and disrespect (see: Priene Inscription).

Christians also defied prevailing attitudes about Roman military service. There were many who refused to serve at all, and those that did were faced with discrimination and conflicts of conscience due to mandatory attendance at official sacrifices. On the other hand, the funerary inscription of Aurelius Mannos, from Phrygia, showed he was proud of both his Christian faith and career as a soldier.

It's a misconception that Christians offended Romans through abundant proselytizing or soliciting converts, or because they led lives of austere simplicity that contrasted unfavorably with the luxury and decadence surrounding them: after the events recorded in the New Testament, there's little evidence that Christians were as publicly active, except in rare cases or when severely persecuted. What really drove negative opinion of them was their self-isolation and secretiveness within Roman society, while this makes sense from the Christian belief system at the time. Christians were noted for largely avoiding public baths. The impact of this is bigger than you may think, since the baths were massive centers of social and city life, and a subset of people that disdainfully refused to participate would have engendered hostility from other Roman subjects. Christians also might have been less sanitary than the general public because of it!

Christians also jealously guarded their rites, especially the Eucharist and Baptism. The New Testament contains descriptions of these events, but no real record or process for how the believers were performing them: Christian talk of "love-feasts" and eating body and blood led many Romans (not unreasonably, I'd say) to conclude that Christian meetings were incestual or cannabalistic orgies! This might sound ridiculous with all we know of modern Christian worship, but remember that Rome was a place of varied beliefs and eccentric, violent practices like animal sacrifice and self-mutilation, so was this really out of the question for an unwashed band of weirdos? Others assumed that the quick rate of Christian converts was partially caused by their use of erotic magic, to seduce spouses from the beliefs and relationships of their partners.

All of these differences led to negative popular sentiment and even anti-Christian mobs, and from the perspective of the Roman authorities, the Christians were the instigators of this unrest. And if there's one thing Rome had consistently proved to be intolerant of, it was those who willingly caused civil instability. Little wonder that the authorities came down so hard on Christians when they intervened.

tl,dr; Early Christians intentionally bilked Roman society by going against longstanding societal norms, though not the ones you may expect (excessive preaching/converting). They would also have been seen as antisocial and probably smelled bad.

Sources: A History of Christianity, Diarmaid McCulloch: p.109, 155-160