First of all, we have to distinguish between Puritans like Oliver Cromwell and other Protestants who were much less anti-Catholic. Both groups disliked Catholics, but the Puritans did so more virulently and for more reasons than the members of the Church of England (for example) did.
The Puritans' dislike of the Catholic Church was mostly religious. The Puritans hated the hierarchy present in Catholicism, its ritual nature, and its idea of people gaining salvation through their actions. They believed in a much more egalitarian church with simple church buildings and services and they believed that people could only be saved through God's grace. They felt that England needed to get as far as possible from these aspects of Catholicism so that God would not punish them. This led them to be very angry even when things like Archbishop Laud's attempts to introduce more ritual into the Church of England.
Other Protestants were opposed to Catholic rule more for political reasons (which mattered to the Puritans as well). Catholicism was associated with enemies of Great Britain such as Spain. They did not want a situation in which Catholics who might be sympathetic to Spain ruled England.