How do the American Romanticism writing styles/values differ from the Enlightenment writing styles/values?

The values of the Enlightenment, also sometimes referred to as the Age of Reason, included a focus on rational thought and logic. Enlightenment thinkers privileged the scientific method and reason over imagination, fantasy, or ideas about the supernatural. They were less concerned with the human spirit and more concerned with what is observable, with what is tangible and ultimately knowable about the human experience. Romantic thinkers, on the other hand, privileged emotion over reason, believing that the ability to feel deeply doesn't have to be taught, and so this makes it more fundamental to the human experience. They were interested in individualism and our own creative potential, and they much preferred folk and fairy tales or legends, as well as folk art forms, over anything that had the appearance of sophistication or polish. Often, there is a greater focus on nature in their works, as well as how nature's sublimity can have an uplifting or purifying effect on us and even improve us morally. One is also much more likely to find references to or possibilities of the supernatural and other fantastic elements in Romantic writings.
In a significant number of his stories, Washington Irving creates some fantastic or supernatural elements, and because they feature American settings (rather than European), they became instrumental in developing a folklore for the new United States. He was particularly interested in capturing and representing the local color—and sometimes the superstitions—of his home, the Hudson River Valley in New York state. For these reasons, then, he is associated with American Romanticism. His works contain everything from people making deals with the devil to superstitious townsfolk telling local legends about a headless horseman who haunts the woods around Tarrytown. Further, attention to nature can be found in many of Irving's stories as well. One need only read the first few lines of "Rip Van Winkle" for an example. The narrator says,

Whoever has made a voyage up the Hudson must remember the Kaatskill mountains. They are a dismembered branch of the great Appalachian family, and are seen away to the west of the river, swelling up to a noble height, and lording it over the surrounding country. Every change of season, every change of weather, indeed, every hour of the day produces some change in the magical hues and shapes of these mountains

This sort of attention to the natural, physical setting of the story is quite characteristic of Romantic writings. Notice that Irving even personifies the Catskill Mountains, suggesting that they "lord over" the country, boastful of their nobility and height. He calls them "magical," and later "fairy mountains," strongly implying that there is something otherworldly, something supernatural even, in their beauty. The narrator also remarks on the effect these mountains have on the "good wives" who live in their shadow. We can see how he is creating a new American folklore, a tradition associated with American Romanticism, just from these few lines and the choice words Irving uses.