They were pure Calvinists. They believed in 1. the Total depravity of man - all have sinned and all human beings will continue to sin throughout their lives and all things that we do is all tinged with sin. 2. Unconditional grace offered to the elect (the chosen few). That is, no matter how many times the elect sin, or what they do, they will be a member of the elect. 3. Limited atonement- that is, only some people (the elect) will receive the benefits of Christ's atonement for sin on the cross, the rest of humanity would perish. 4. Irresistible grace - the elect can never lose their chosen status, again, no matter how much they sin. 5. Perseverance of the Saints - you cannot ever lose your election.
Most Protestants don't agree with the idea that you cannot lose salvation or that there is nothing we can do to earn it.
Most Catholics believe that St. Peter and his fellow priests are the ones who convey salvation through the sacraments, and that you have to pay for your sins through confession and atonement, something not necessary in Puritanical theology.
Incidentally, the Puritans were not all that pure. They mostly lived in one-room houses, cussed and drank and regularly had sex in front of their children. (How else to have a second one?)