It is used to express connection with or discontinuity from the "old" Rome, depending upon context, and is particularly used by the Greek Orthodox Church to emphasise that the see of Constantinople should be considered as second only to Rome in prestige.
It has been a cultural, historical, and theological concept within much of European culture (as far east as Russia) for centuries if not millennia.
Paris has at various stages of its history been designated "nouvelle Rome" or New Rome, as early as the reign of Philip IV (1268-1314) but from a tradition starting most significantly under the rule of Louis XIV who dominated most of Western Europe, and whose capital experienced massive increases in population, wealth, lavish royal building projects (there were 500,000 people in Paris by the mid-17th century, compared to 350,000 in London). However it was Napoleon III's appointment of Baron Haussmann as city planner of Paris in the mid-19th century, that is the cause of the appellation in modern times.
Within the context of Protestant Reformation, it became a pejorative description, applied to nations or cities that earned a reputation for rapacity, immorality, or other social or political faults. This may have its roots in virulently anti-Roman(Catholic) propaganda against "papists" and the city of Rome, home of the Pope and heart of the Roman Catholic Church, which drew the ire of many a Reformation author. In the present day, "New Rome" is used in this form mostly to refer to "political immorality", casting any large and powerful country into the role of an oppressive and expansionistic empire. "Babylon" is often used in a similar sense.