Both types of writing reflect the culture and beliefs of their era.
Writing of the Puritans particularly captures the lifestyle of a people bound to the conditions of a strict religious view with little room for grace or forgiveness. They focused on education, and founded Harvard University among others. Their literature reflects this in that the language demonstrates complicated (but correct) sentence structures as well as great variety in vocabulary. I think it is also important not to confuse writings about Puritans (The Scarlet Letter, The Crucible) with writings by Puritans.
Anne Bradstreet's writings reflect someone feeling trapped by a society's pressure to perform or be perfect. Jonathan Edwards' sermons reflect great pressure to come to Christ. Each of their writings use literary devices to express their points. The development of these are usually in the form of a more complicated extended metaphor or allegorical symbol. We see this in the hand of God in Edwards' "Sinners in the Hands of an Angry God" and in Bradstreet's "The Author to Her Book" in which Bradstreet likens her writing to that of raising a child.
In the Romantic era, we see something much different. In essence, Puritans wrote about what was real to them; Romantics began to express themselves through their imagination. A wide variety of authors are considered to be Romantic. Mary Shelley for example wrote Frankenstein which was a great expression of what the human mind was capable of imagining at the time. Jane Austen and the Bronte sisters were also considered Romantics. They often wrote about relationships which contained exaggerated situational circumstances.
There are a number of differences between the two eras, but the main one is that Puritan writing was entirely focused on God and religion. Writing was a means of servig God. Take the poetry of Edward Taylor, for instance. His poems were a means of making God more available to the masses. Even the histroical journal of William Bradford, "Of Plymouth Plantation" contains many incidents of praise to God for safety and a new home in the new world.
The Romantics looked at the world differently. It was a reaction against Realism and an escape form a rapidly industrialized world, Great value was placed on intuition (common sense). The dark romantics such as Poe began to delve deep into the human psyche as well.