Caio Mussolini, one of Mussolini's grandsons stood for office in 2019, while his cousin, one of Mussolini's granddaughters, Alessandra Mussolini was elected to office and served for years. [Rachelle Mussolini, another granddaughter](https://en.wikipedia.org/wiki/Rachele_Mussolini_(politician)), is currently a councillor in Rome. All three are considered "far right-wing" in Italy and are not shy about mentioning their relation to the fascist dictator.
Despite my living a stone's throw from the Italian border, I confess I don't know much about the country or its culture. What is it that allows Mussolinis to openly campaign with the surname of Mussolini, while the name "Hitler" is wildly vilified and even the first name of Adolf has seen a sharp drop-off in Germany after WWII (site in Germany, but you can't mistake the graph).
Update: Obviously, that last question about "openly campaign" falls afoul of the 20 year rule, so I withdraw it and would stick with the question in the title.
First, there is a very simple answer: Hitler had no children. His sister Paula also died without descendants. His half-siblings Alois jr. and Angela had children, but Angela's children obviously had her husband's surname, and Alois jr.'s only surviving male child also changed his surname to "Stuart-Houston." In any case, none of these blood relatives of Hitler had children and now the remaining ones are very old.
Conversely, direct descendants of Mussolini there are still several. The bald man had two sons who survived the war, Vittorio and Romano. Also a nephew of his, Vito (son of Arnaldo, Benito's brother), survived. Should we consider the descendants of the daughters, who have the surnames of their husbands, the list would be even longer. And some are very young, if not yet children.
But there is also a deeper discourse to be made, as you rightly point out. Even if he existed, no Adolf Hitler Jr. could ever enter the Bundestag and at most he would be the revered fetish of some neo-Nazi organization.
After World War II, there was a need to reintroduce Germany and Italy into the community of Western European states, both to facilitate the economic reconstruction of the continent and to organize military defense against a possible Soviet invasion. Obviously, public opinion in the invaded countries was not at all happy to see the two former Axis powers rearm. So an intensive "public relations" operation was made to separate the now democratic Italy and Germany from their past. In Germany all the blame for the war and the Shoah was placed on Hitler, the Nazi regime and the SS. Thus was born the myth of the "good Wehrmacht," in order to make the rebuilt Bundeswehr acceptable by reflex, and even the Shoah itself was ignored as far as possible by those who witnessed it firsthand. Current German guilt is much more the child of the 1960s protest than of Nuremberg.
Something similar happened in Italy, but ... there, too, Hitler and the Nazis were blamed for everything! Mussolini's co-responsibility in the war and extermination goes out the window, and the predominant view in Italy is that it was a country invaded by Germany on a par with the rest of Europe. It is a narrative that suits everyone: to the political forces heirs of the liberation war who can focus on the part that saw them first heroically resist and then repel the Nazi invader; to the Armed Forces who can forget about a page shameful for failures and war crimes; to the nostalgic of the regime who can hide the bloodiest parts of it.
Another point is that 1945 is for Germany a true "Stunde null," a "zero hour" in which everything that existed before was no more. Germany itself ceases to exist as a united country, and the two successive Germanies born years after the conflict did not have the same name nor the same institutions nor the same codes as the Reich born in 1871. Scholz does not hold the same office as Hitler: one is Bundeskanzler (of the Bundesrepublik Deutschland) and the other was Reichskanzler (of the Deutsches Reich). Italy, on the other hand, maintained institutional continuity, so it was more difficult to expunge what Fascism had put into it during a period of strong modernization. For example, the Italian penal code, although much modified, is still the Rocco Code written in 1930. And Mussolini was officially "Presidente del Consiglio dei Ministri" as much as Meloni will be in a few days.
Today, only for politically active people on the left (and not even all of them) Mussolini is the dictator of racial laws, concentration camps in Libya and Slovenia, mass rapes in Greece, unprovoked attacks on neutral countries, poison gas on civilians, active collaboration (during the puppet regime 1943-1945) in round-ups and deportations of Jews, political opponents beaten to death by the hundreds if not thousands.
Most Italians today remember Mussolini as a dictator, but at worst as an inoffensive waffler who behaved normally if not well in many aspects until the fatal mistake of allying with Hitler.
Translated with www.DeepL.com/Translator (free version)
While waiting for an answer on this exact subject, you may be interested in reading this answer about the ideological differences between the two regimes which may shed some light regarding the modern perceptions of their leaders’ descendants.