What are five legacies of Puritanism that influence us in 2018?

Many aspects of the Puritan worldview affect us today. I will answer this question from an American perspective, since it is in this country that Puritanism remains most influential.
The first is directly related to the idea of American exceptionalism. The puritan John Winthrop, in an address to the early settlers at Massachusetts Bay, exhorted them to be as a "city on a hill." He meant that the nature of their religious experiment made them different, but this idea has made its way into American politics and culture. Many politicians, ranging across the spectrum from Ronald Reagan to Barack Obama, have repeated the claim that the United States is a divinely-chosen nation that serves as an example to the rest of the world.
Another political aspect of Puritanism, at least as it was established here in North America, that remains today is the concept of participatory government. Puritan towns featured a "town meeting" in which the people (i.e., church members) made important decisions that affected everyone. These included naming ministers, constructing town buildings, and so on. Many American towns still have this form of government, and the idea of civic engagement still permeates American political culture.
Long ago, a sociologist named Max Weber identified a "Protestant [by which he basically meant Calvinist or Puritan] work ethic." This was the idea that a person could glorify God through his work and that making money was not inherently sinful. This encouraged people to work hard and to invest their money in smart ways. American culture continues to celebrate work for the sake of financial gain, and, in many ways, associates work with religious virtue. In no other country are religion (especially Protestantism) and economic philosophy (i.e., capitalism) so intertwined.
Another way in which Puritanism remains influential is in its links between sexual behavior and morality. While Puritans were never as prudish as portrayed in popular memory, they certainly placed an emphasis on sexual behavior as a marker for one's virtue. This was especially true of female sexual behavior. While sex is very prominent in American popular culture, it is indisputable that modern mainstream evangelicals place a heavy emphasis on sexual behavior, especially outside of marriage.
Finally, as the above answers have suggested, the ideas of Puritanism, especially the concept of a "fallen" world in which people were in need of God's mercy and grace, are fundamental to many branches of modern evangelical Christianity. Evangelicalism remains a powerful cultural, political, and religious force in American society, with many Puritan values at its heart.
https://www.nytimes.com/2012/08/05/opinion/sunday/are-americans-still-puritan.html