Puritanism was a radical challenge to the Roman Catholic Church in at least two ways. It was a radical challenge in its doctrines about salvation and it was a radical challenge in its doctrine about church hierarchy.
The Catholic Church held that people could, in essence, earn their way into heaven. That is, people could do enough good acts, and enough penance for their bad acts, that they would deserve a place in heaven. This was why the Church had sacraments like confession/reconciliation and policies like the issuing of indulgences. By contrast, Puritans said that no human being could earn salvation. Humans were so bad that they all deserved to go to hell. Moreover, people were predestined to go the heaven or hell and nothing they did could change that. Thus, the Puritans attacked the very basis of Catholic practice, saying that Catholic sacraments and practices could not earn a person salvation.
The Catholic Church was very hierarchical. Priests are supervised by bishops. Bishops have their own supervisors. The whole church is ruled by the pope in Rome. What the pope says (in certain circumstances) is law for the whole church and everyone must obey. Puritans did not believe that this hierarchy was what God wanted. They believed that no one person had the right to tell anyone else what to think and how to worship. Therefore, Puritanism was very decentralized. Each pastor was essentially autonomous, answering to no one. This undermined the Catholic idea of centralized control under a pope selected by God.
In these ways, Puritanism was a radical challenge to Catholic teaching about salvation and to the structure of the Catholic Church.