The institutional structure of the Church was far more sophisticated than in the late Roman Empire, when it was in may ways still a grass-roots movement. Also, the Church was far more doctrinally consistent during the High Middle Ages. There were many different variants of the faith that modern-day Christians probably wouldn't recognize as Christianity floating around the Mediterranean basin during late Roman times.
Generally, the power of the Church expanded both in terms of money and political power. People would donate money to the Church so their sins would be forgiven. They would do things like going on Crusades for the same reason. This, plus the fact that the Church owned a lot of land (and even had control of the Papal States in Italy) meant the Church's power was growing.