First, it's important to recognize that in most cases it took centuries, if not a full millennium, for these traditions to be fully uprooted.
Christianity had a few things going for it as a missionary religion that the divergent local pagan traditions couldn't match: it came out of Roman late antiquity with a strong and international institutional identity and organization; it was better able to integrate itself into both the local and royal political orders; and it demanded exclusive worship of its followers, which sapped the support of other traditions. And apparently the beliefs were appealing to people from a wide variety of backgrounds (never underestimate the role of either women or desperate royals looking for political backing in the spread of Christianity)
The pagans, meanwhile, were mostly just local communities worshiping in their traditional manner.
This is possibly a terrible analogy, but each branch of paganism is somewhat like a bunch of locally owned restaurants run by individual owners, all serving similar cuisines, who couldn't compete when the giant Mediterranean franchise (under a regional manager called a bishop) rolled in and took their lunch-money, all with the support of the government.
For a different, but excellent perspective, check out this recent thread:
http://www.reddit.com/r/AskHistorians/comments/38o4wb/did_pockets_of_paganism_survive_in_medieval/
The last group of pagans lasted up until the 14th century. The Baltic Pagans managed to exist so long mainly thanks to isolation and generally keeping to themselves. However, Prussian raiding groups eventually led to a Polish Prince calling upon the Teutonic Order to come and cleanse the pagans. Within the next few decades the Teutons preform something that can best be described as religious genocide, mercilessly slaughtering pagans as they went. Eventually, the Grand Duke of Lithuania, Jogaila, married the 14 year old Queen Hedwig of Poland and converted to Catholicism in an attempt to stop the war with the Teutons. The Teutons continued until their decisive defeat at Tannenberg in 1410.
Firstly, it's incorrectly to think of those as religions. They're just names to refer to a collection of similar beliefs (similar religously, culturally, and linguistically).
The Hellenic and Roman religions were the closest any of the pre-Christian faiths got to organized religion and both of them were collections of massive pantheons with many regional gods.
Christianity on the other hand was a single religious faith with getting others to follow their faith being a core aspect of their belief. Christianity had multiple sects with varying views of what should or should not be orthodox (with the two famous examples being Catholicism and Orthodoxy, which, funny enough, both mean universal/true in their own way) but they would compete to convert further as opposed to just squable.
The conversion of Great Moravia is a good example. It began being converted by Greek (not yet Orthodox) Christians but Latin (not yet Catholic) Christians arrived and managed to gain favor in Great Moravia.
The missionaries and evangelists have been in the Indian subcontinent since the 4th century. Yet, there are only 2.3% Christians in India.
Christianity had the advantage of having an universal god independent of location and tribe, which was a novelty in thinking. You find such evidence in the Bible. In addition was Roman culture popular and this gives a strong influence.
In the roman Empire itself early Christians became popular because of their work for the poor. Already the Jewish religion became very popular because of the concept of a universal god and so became Christianity popular too.
They did not. I live in a Slavic country, the Republic of Macedonia, and Slavic 'rodnovery' is extremely prevalent here. Even though the country thinks of itself as demochristian, the truth is that worship of Yarog, Svarog, Dazhbog and the others is open and apparent, magicians and . The Orthodox Christian church has been fighting it for decades, and magicians and other figures are prominent.