What is an important similarity between the way scientists were thinking during the Scientific Revolution and the way philosophers were thinking during the Enlightenment?

The Scientific Revolution paved the way for the Enlightenment. The Scientific Revolution began between the mid-sixteenth to mid-seventeenth century. Some historians use Nicolas Copernicus's heliocentric model of the universe, and others use Galileo's "Sidereus Nuncius" ("Starry Messenger," in which he published his findings from his views in a telescope) to mark the beginning of this period. These publications are separated by about sixty years (published in 1543 and 1610, respectively).
The major linchpins of the Scientific Revolution were Copernicus, Kepler, Galileo, and Descartes. Copernicus's brainchild was the model of the universe with the sun as the center (anathema to the conservative Church, which required that the Earth be the center of the cosmos). Johannes Kepler (working in the Copernican tradition) developed three planetary laws (essentially accounting for their elliptical path around the sun, their varying velocities, and the relationship of each planet's orbital period to its distance from the sun). Galileo Galilei was the first to use a "spyglass" (i.e., telescope) to observe Jupiter's moons, the craters on the Moon, and the general existence of many more stars than previously thought to exist in the cosmos. Galileo was the most combative of his colleagues and wrote a treatise ("Dialogue on the Two Chief Systems of the World") in which he championed the Copernican model of the universe. This evoked the ire of the Church, and Galileo was prevented from publishing further by the Inquisition. Descartes, born in Catholic France, moved to the Protestant Netherlands after Galileo's condemnation. His chief contributions to the Scientific Revolution involved investigations into the nature of the soul. He promoted deductive (general to specific) reasoning to make the famous proposition "I think, therefore I am," as well as to claim that the universe comprised only matter and motion.
The Enlightenment, conventionally understood as beginning with the publications of Isaac Newton (viz. "Principia Mathematica," 1986) and ending as late as Napoleon (d. 1821), was indeed concerned with reason but less focused on the natural sciences and more on reason's ability to negotiate the relationship between the individual and his natural government. The major players of the Enlightenment included John Locke, Thomas Hobbes, Adam Smith, and Jean-Jacques Rousseau. Locke believed that an ideal government should settle disputes impartially, as well as protect people's rights to property. He championed the necessity of the consent of the governed. Thomas Hobbes argued that the natural state of a human being was warlike and that an effective government needed to inspire a fear of death in its subjects. Government, according to Hobbes, must have total authority. The Scottish economist Adam Smith proposed a laissez-faire capitalist economy, in recognition of the fact that man was self-interested. Self-interest (according to Adam Smith in an economic sense, as well as to the other political theorists in a civic sense) is a natural law. Rousseau shared Locke's views of the necessary consent of the governed; however, he diverged from Hobbes in his appraisal that humans were inherently good. It was the government's primary duty, according to Roussseau, to promote the will of the people.
As a total movement, the Enlightenment claimed that human laws should be based on reason, rather than conservative tradition. The Scientific Revolution, too, highlighted reason and careful observation. While Galileo was somewhat of a maverick, the Scientific Revolution thinkers had no apparent social axe to grind. That said, the overarching focus on reason (as opposed to faith or observation alone) is the chief similarity between the early scientists of the Scientific Revolution and the thinkers of the Enlightenment.
Just as the Scientific Revolution paved the way for the Enlightenment, the Enlightenment would inaugurate nineteenth-century Romanticism (concerned primarily with man's relationship with nature).