I'll go with Souron on this one. No, it wasn't the money and power that made it spread, well at least not the early church. Quite the opposite I'd say. Christianity embraced the sick, the poor, the weak, people of the female persuasion (two kinds in ancient Rome, wives and whores), slaves etc. More importantly, the early leaders of the church knew how to organise. So they get these nobodies to pull together, to help each other. There has been so much insincere ramblings about Christian love, and the Christian thing in helping thy neighbour, that people tend to forget that this was what the first Christians did in a big way. Part of the success was the fact that they filled vital social needs for all the little people that the ruling segment of soceity couln'd have cared less about. If you were a non-citizen commoner in the Roman empire you were pretty much free to die the best way you could. So they had to help each other, and Christianity became the most succesful framework to do that within.
Rome (and Ancient Greece) were based on pretty harsh aristocratic ideals. (Such as self-control and the extermination with extreme prejudice of all political opponents.) The Roman and the Greek ruling segment of society held that things such as compassion and pity were very base feelings, they were despicable "turpissimus" to the Romans. And in order to educate the public you get gladiatorial games, so that they will find blodshed, pain, mutilation etc. to be acceptable aspects of life. Mercy, they could understand, but only as something to be granted to an opponent (of equal standing) for political reasons. (I.e. his family/country etc. might do you a dirty one otherwise.) God help you if you were a rebellious subject.
Enter the Christians; these guys have inverted all these ideals. The forgive their enemies, help each other, actually care for the little folk, and face death in its most agonizing forms without flinching. (Well, that was what one expected of any Roman patrician, but finding this kind of fortitude in common slaves... well, it was kind of disquieting, sort of indicated that the social order was maybe not as natural as everone had assumed.)
So Christianity began as a grass root popular movement better att organising its members for mutual support compared with competing religions, and eventually it could challenge the established ideals of the political elite and win them over. After Constantine you get the pagan emperor Julian the Apostate, who tried to suppress Christianity and bring back the worship of the old gods. Still, Julian had to admit defeat in a sense, since his reforms of the ancient religion used the way the Christians had organised themselves as a blueprint. He actually saw the necessity of fulfilling the social function Christianity filled. It was by then impossible to go back to the old ways of lording it over slaves.
And then Rome becomes Christian... Glory be, thinks all right thinking, church going Christians, from now on all should be well. But pretty soon you get all the kind of nasty problems you always get when combining secular and religious power...