How did the Christian take over?

I can see how it follows logically that since the Americas (and Australia) were primarily colonized by Western Europe that their religion would dominate in those areas, but how did it come to take over Western Europe so thoroughly in the first place? What is it about the religion that was so appealing that people converted? Or was it simply the advantage of having the same religion as the emperor (meaning Constantine, since he seems to be the lynchpin of popularity, from what I've read)? Were there tangible (monetary/civic) advantages to being Christian in the late Roman Empire?