This is a complex question and not one admitting of an easy answer. I'm going to give you a number of factors to consider.

Firstly, there were a number of features of Christianity that made it unattractive to Roman society.

1. Judaism was seen very much as an ethnic-religion within the rest of the Empire. It's place was tolerated based partly on its antiquity (remember that for the Romans 'in general' antiquity was revered, novelty was not something to be prized) and partly on the very strong association of the religion with the ethno-nation of Israel. Judaism in the 1st century, I would argue, was not very syncretistic and not very adaptable to Roman practices, and so maintained its independence whereas other, polytheistic local religious practices were happily incorporated into Hellenistic and Roman religious life.

2. From this perspective, few Gentiles converted to Judaism. Judaism did not practise wide-scale proselytism and the emergence of the category of Gentile 'god-fearer' as a virtual permanent state of being rather than a preliminary one to conversion in part bears this out.

3. Christianity, at least initially, was seen in terms of its Jewish origin. It differed in that it became distinctly not-ethnic in terms of membership. This, as Christianity became better known, actually harmed perception of Christianity since now it was (a) no longer tied to an ethnicity, (b) novel instead of ancient.

4. However, at the same time the move to include Gentiles without requiring them to become Jews, the struggle and outcome of debates within 1st century Christianity, opened up Christianity for the ready inclusion of Gentiles and meant wide-spread conversion was possible.

5. Furthermore, unlike 1st century Judaism, Christianity was significantly pro-active in proselytism in a way that Judaism wasn't. It was actively concerned to win converts, Jewish and Gentile.

6. Overall, however, for at least the first 3 centuries, to convert to Christianity was socially disadvantageous.

Now we should consider how Christianity developed and spread through the Roman Empire to a point where it became dominant.

1. Embrace of Gentiles. Christianity, in contrast to Judaism, readily accepted Gentiles without change of ethnic identity. It succeeded in divorcing ethnic and religious identity.

2. Pratices of social welfare. In a well-known quotation from the Emperor Julian, he laments that pagans are not more socially active in caring for those in need. Practices of early Christian charity outstripped their neighbours and probably did a lot to improve social perceptions of the Christian religion.

3. Practices of martyrdom. While persecution of Christians in the first three centuries was generally sporadic and localised, it certainly did occur, and towards the end of the 3rd century it happened on a broader scale. And while also we should acknowledge that Christian martyrdom was not always the outcome, that plenty of believers capitulated under pressure, the overall impression from sources is that the maintenance of Christian belief unto death caused a general respect for Christian beliefs.

4. Religious alternative. Against a background of Roman pagan religiousity, which rested on notions of duty, contractual obligation, societal norms, and Hellenistic mystery religions, Christianity offered a religious alternative that, to many, was compelling. The strengths of Judaistic monotheism, moral practices that contrasted in many cases with prevailing Roman norms, an apocalyptic worldview, an alternative to transactional models of religious influence, and a fairly personalised relationship with divinity.

5. There was a tipping point where Christianity moved from margin to centre, but its definitely complicated; by most accounts, Christian population in terms of percentage of Empire at the conversion of Constantine was probably not more than 10%. It was still officially being persecuted. However we parse out the events of Constantine's conversion experience, it changed the playing field by situating a powerful figure as a Christian, and moving Christianity from proscribed to tolerated, and eventually to established. This massively changed the social consequences of being Christian from disadvantage to social advantage, and you see considerable conversion through the fourth century that moves Christianity to the majority religion.

That's my introduction to this field. I teach courses in Christian history and Christian origins, and once I get home I'll edit this with major secondary sources on the topic.

I'll just add here, in response to the suggestion that it was Christian 'adaptability' that the whole question of 'adaptation' needs to be fleshed out before it can be argued. Specifically, Christianity does show measures of cultural adaptability, but we need to consider more carefully what practices were adapted to or from pagan religions, in specifics not in generalisations, before we can make any judgments on this. Personally I think merely saying "Christianity embraced elements of paganism in order to be more acceptable" is vague and unproven. We must inquire "what elements, from which religions, and how incorporated?"

/u/MarcusDohrelius would probably be more equipped to answer this question, but until they show up, I'll take a crack at it.

First, we have to be careful not to think that Christian conversion was a quick or painless process, even after Constantine's conversion or the Edict of Thessalonica. Paganism continued to exist as did myriad forms of Christianity aside from the officially sanctioned creed. It took centuries for Christianity to become ubiquitous, and even longer for any sort of unity among the Christian sects to become the norm.

As to the answer of what was so appealing about Christianity? There are three basic answers to this. First, the practical aspects of the creed should not be underestimated. Here was a religion that promised salvation and hope and when one considers the environment of late Antiquity, it's easy to see the appeal of that message. Second, with the support of the Imperial apparatuses, Christianity became a route through which men of status could gain access to the levers of power and influence, either directly in the Imperial regime, or through the various Church positions. Bishops were both important eccleciastical and temporal positions. It is also easy to see why the Imperial administration of the late Empire would find adopting an 'official' religion as beneficial, as it gave them another instrument with which to effect control over the Empire. Remember, this was during a period when the Emperor had long since ceased being the princeps and was now a highly ritualized, almost otherworldly figure. Thirdly, the Christian leadership themselves showed a great deal of practicality when adopting Christian dogma to existing pagan beliefs across the empire. Many Christian traditions that seem so different from Judaism actually arose because of the adoption of the practices and traditions of popular sects that already existed, like the cult of Mithras in the west. This was not uncommon for the Romans obviously, as adoption of gods into their Pantheon seems to have been relatively common, but it did aid in the acceptance of Christianity.

As to whether it replaced anything, the answer is probably best described as co-opting. The Roman elite flocked to whatever would give them the best avenues to power, whether that was being appointed the high priest of Apollo or a Bishop. Is there much difference between a Temple to Athena that draws visitors and donations and a Monastery that claims to hold relics of a saint that does the same? The only real large change I would likely agree on is the fact that the imposition from above more closely connected the Imperial and religious posts, as opposed to the largely disjointed and localized control over religion that existed prior to this.

EDIT: Forgot the source, which for me was Christianity: The First Three Thousand Years by Diarmaid MacCulloch