What were the Puritans beliefs compared to Jamestown's?
You've already gotten an excellent answer. Let me just add a few more tidbits.
As a whole, the Jamestown settlers were, in fact, only there for economic gain, despite their claim that they were also interested in "converting" the "Savages" (Indians). Only one member of those original settlers was a preacher, and there were no women in the bunch. That's a pretty clear indication that these men came to grab what they could--as they had been sold on the idea of a "land of riches"--and head back home. They had no intention of settling or staying, as is also evident by their lack of foresight in bringing along enough food to sustain them for any length of time.
The Puritans, on the other hand, were a mixture of men and women, nearly all committed over the long term (both in England and in Leyden, Holland, from which they departed for America) to the principles of faith, salvation, and sharing the "good news" with the natives. They weren't persecuted for their faith in Leyden (though their lives were difficult for other reasons), but they believed they were to come to America to promote the gospel and evangelize from there to the rest of the world.
In terms of this novel, the Puritans of 1692 were rather different than those first settlers in one important way. The first group to arrive underwent hardships of every kind and relied on God for everything--literally everything. They had little esle on which to rely. They were a spiritual people and it showed in everything they said, built, wrote, and did. Generations later, life had become much easier and they weren't as reliant on God as Provider. Instead, they saw Him more as either Savior or Punisher--and one could not know which until Judgment Day. Thus their desire to "root out" sin in an effort to prepare for that day. In essence, they took over the role of judge here on earth, meting out punishment for sin without any forgiveness of sin.
This skewed or partial view of God and sin and punishment was, of course, part of the atmosphere which led to the witch trials in Salem.
The Puritans and Jamestown individuals held some fundamentally different starting points on the founding of their respective colonies. Jamestown was primarily a financial endeavor. A British company, the London Company, took out a charter to establish the colony. This helps to underscore that the primary rationale for discovery was economic, to help make the company money. This economic rationale was tied into the discovery of tobacco and the exporting of crops back to England. Religion was present, but the drive for expansion of an economic base that would benefit England was the primary level of focus. This was not so in the Puritans' discovery, where the desire for religious freedom animated and initiated the movement from England. The belief in original sin, and the idea that human beings were, essentially, "sinners in the hands of an angry god" was felt throughout Puritan society. The emphasis on materialism and economic growth that was an embedded part of the Jamestown society was something not as felt in its Puritan counterpart. The presence of God and the hope of reclaiming a spiritual identity was something that drove the Puritan's theocracy, a political order where religion and government was not separate, but convergent. As a result, Miller's work correctly identifies one of the major problems when the line between religious zeal and political justice becomes blurred.