Fundamentally, I think this can be explained simply by the major theological differences between the different traditions. As you have framed the question by putting Judaism in relation to Christianity I will answer it in this way. But of course if you asked 'What were the reasons that Christianity spread throughout the Roman Empire?', the answer would be much more complex.

Put simply, Judaism is not and has never been a religion that teaches its followers to 'spread' the religion through proselytisation. Missionary activity and conversion are, on the contrary, quite taboo, and there are a ton of issues around what it means to be Jewish, and who can 'become' Jewish. Theologically speaking, this aversion to proselytisation emerges from the fact that when God made his covenant with Moses and the Israelites on Mount Sinai, he gave them and only them a special role to play in the world. Jews, the descendants of those people, have thus been 'chosen' to follow a special set of rules and observances which are dictated throughout the Tanakh, whilst 'gentiles' or non-Jews are not required to keep these; nor should they, for God did not make this special covenant with them. The stories of the Hebrew Bible are about Israelite kings and leaders, Israelite experiences and memories of exile, punishment and turmoil, and tales of God helping the Israelites regain their 'promised land' in the Canaan through the destruction of their enemies. It is quite fundamentally a religion that is based around the specific experiences of a specific group of people, and is not therefore geared towards the conversion of all of humanity in order that they might attain salvation. I am no expert of Judaism, but I have read what the Hebrew Bible and studied some Jewish theology and as far as I am aware there is a place for gentiles in the afterlife so long as they follow the seven laws that were dictated to Noah by God, and would therefore be righteous and just. They do and did not need to convert to Judaism, practice Jewish rituals, observe Jewish holidays, follow Jewish laws etc in order to achieve salvation.

Christianity, of course, emerged out of Judaism as a messianic sect. Its prophet Jesus Christ expressed the idea that in order to achieve salvation one must simply have faith in God, or at least this was how early Christians interpreted his teachings. Christians generally believe that at the Last Supper, God made a New Covenant with humanity through the Eucharist, as well as a New Commandment, to love one another like Christ loved humankind. Indeed, many of the stories and parables of the New Testament are about loving one's fellow man, tolerating those that are different from us, and uniting in harmony under one God. It is therefore a religion philosophically inclined towards influencing others, and littered through the New Testament are commandments to go forth and 'make disciples of all nations', to spread the good news that Christ the messiah has died for our sins. If we look at the origins of Christianity, in figures like John the Baptist, and the needs of 'God-fearing' peoples (Gentiles living in Judea who combined their pagan Hellenistic religion with aspects of Jewish ritual and worship), this tendency towards proselytisation running throughout the Christian faith is perhaps unsurprising, although we could obviously go into the social and political context of Christianity and the many complex reasons for it's emphasis upon its followers spreading the gospel in much more detail than this.

Importantly, In the early years of Christianity a central debate was waged over the issue of proselytisation. Early Christians were Jews who had accepted Christ as Messiah and therefore were essentially considered heretics by the mainstream Jewish authorities (although again we must remember that before the destruction of the Second Temple what we might call 'Judaism' was much more theologically diverse than now). Christians argued that their beliefs were justified via this New Covenant with God; but did the New Covenant completely supersede the Old? Should Christians, who again were mostly 'heretical' Jews, keep to the commandments and observations that are described in the Hebrew Bible? And should Christians be proselytising to so-called gentiles, given that in order to achieve salvation, one must follow the example of Christ's grace and have faith in the Lord? Paul the Apostle played a central role in overcoming this debate in the mid-1st century AD. In these early years of Christianity, Christian sects like the Nazarenes preached to Jews and gentiles alike, but in order for gentiles to convert to their Christian faith they had to observe the many laws and rules of the Hebrew Bible. This was an issue for a number of reasons; perhaps the most important one, in light of your question, is that it's actually very difficult for a non-Jew to learn and observe all these rules. Jews have lots of special rituals, holidays, customs, dietry restrictions and the like which of course makes it more difficult for a non-Jew to join the faith, even if they were encouraged to. Perhaps the most problematic for early Christians hoping to convince others to do so is the issue of circumsision, a rite which God in his covenant with the Israelites explicitly stated all Jews must perform. Persuading grown men to remove their foreskins in an era with no anaesthetic and poor hygienic practices surrounding surgical operations was difficult for reasons i'm sure I don't need to overstate, and Paul navigated the debate over the conversion of gentiles in relation to this question specifically. The details are complex and of course what Paul actually meant is subject to a centuries old debate by Protestants and Catholics but he essentially argued that salvation was assured simply by faith in Christ and not by staying true Jewish laws. This is all described in the New Testament, and is the basis of what is called 'supersessionism', the idea that the New Covenant has made the Old Covenant essentially redundant, and that Christians are no longer bound by it.

Whislt fought over technical issues, implicit in these debates are the questions of what it means to be a Christian and not a Jew, and the place of the Mosaic Covenant and the Hebrew bible in Christianity. Before too long, those sects which were once central in Christian society, like the Nazarenes, were sidelined and denounced for continuing to follow Jewish laws. 'Pauline' Christianity tirumphed and allowed Christians to mobilise and proselytise more easily and to a wider audience than just Jews. They had not only the theological will and justification to do so, but the practical advantage of removing the observances of the Hebrew Bible as requirements for joining the faith.

For more information about this complex topic you can consult Wikipedia and read the New Testament. For academic discussions, what is called the 'New Perspective on Paul', a historiographical and theological trend beginning in the 1970s, is very important and a good book might be E.P. Sanders' seminal 'Paul and Palestinian Judaism'.