The European free speech legal infrastructure dates from the ECHR in 1950, and was somewhat imposed by the Allies after the war. As well as a lot of people who wanted to avoid having populist incitement to genocide again.
The limits on free speech in European countries are also much older in parts. The history in Europe on that topic is just very different from the US, I'd say.
Indeed. People forget that e.g. Portugal, Greece, and Spain were dictatorships until the 70s, and that a lot of eastern EU was under Communism until 1990s.
"The EU" is not equivalent to "France and Germany", despite what they think.
The speech culture of France and Germany is very different to the US as well, it's not just the dictatorships of the recent past.
Nazism as an ideology is illegal to promote in Germany. Use of Nazi symbols outside of an artistic or educational context is illegal in Germany and France (most European countries really). Several neo-nazi, neo-fascist, and communist parties have been banned since the founding of the modern German state. Far-right parties in Germany are always walking a fine line - if their membership flirts too much with nazism or fascism they are very much liable to be banned.
Several European countries with strong democratic traditions in the modern era remain very wary of extreme ideologies. Their proponents murdered millions and burned the continent to the ground less than a century ago. Most people are comfortable with the state killing these movements in their infancy.