Though political ideologies have complex histories we can say, in modern America, that Progressive, Liberal, and New Deal agendas are generally about health care for all Americans, less spending on military ventures, more political attention to domestic problems, a strong social safety net for those who are laid off from their jobs or hit with medical emergencies, more democracy (thus, less government control by wealthy individuals), racial equality, and so on.
Nazi and Fascist movements, generally speaking, were (are) about using state power to crush opposition and to dominate other countries through military action. In sum, a philosophy of "might makes right" and violence is necessary or good.
So, why are comparisons between Progressivism and Nazism being made? How, is expanding Medicaid to more low-income Americans similar to the murder of six million Jews? How are calls for a public works program for the long-term unemployed similar to the unprovoked invasion of countries for the purpose of creating a racially "pure" world?
Harry Hopkins, the first and most well-remembered director of the WPA, replied to Herbert Hoover's claim that the New Deal was leading America to Fascism by asking, "Is it dictatorship to try to operate a government for all the people and not just a few? Is it dictatorship to guarantee the deposits of small depositors, and keep phony stocks and bonds off the market? Is it dictatorship to save millions of homes from foreclosure? Is it dictatorship to give a measure of protection to millions who are economically insecure and jobs to other millions who can't find work? Is it dictatorship to try to put a floor under wages and a ceiling over working hours?" ("Hopkins denies relief waste in reply to Hoover on fascism," Washington Post, May 9, 1938.)
Above all, modern Progressivism is a political ideology that holds that government should serve all its citizens, not just the wealthy few; and that government should help its citizens when they fall down (in other words, "We the People" helping "We the People"). It's bizarre that some people think that this type of ideology is Nazi-like. But perhaps such a comparison makes sense, if you happen to be a person who wakes ups every morning wondering how you can make the rich & powerful more rich and more powerful.