So, I think it's all very straightforward. The prefix 'neo' is often attached to terms not because of any major structural or ideological differences, but rather to indicate a revival after an initial rise and fall. 1945 is used by historians to indicate the moment when fascism was defeated on an international level when the Third Reich fell. Some will also go back to 1942, following the German invasion of the Soviet Union, to look at neo-fascism's roots. This is very similar to the way that the term 'neoliberalism' is used. The 'neo' in that sense refers to economic liberalism's revival in the 1970s and 1980s after it became discredited among economics and politicians in the 1930s and 1940s in the wake of the Great Depression. It's all fascism, but the prefix is used to indicate its post-war usage. Some historians may point out differences between historical fascism and neo-fascism, but much of this is negligible.

However, the only major difference is that neo-fascist figures and groups operate in an era when fascism's time as an international political force is well-documented. With fascism having an extremely negative reputation for over seventy-five years, the use of toned-down language or codewords is far more prevalent. For example, because Jews have become generally more accepted into mainstream society, the more successful neo-fascist groups will still produce anti-Semitic content, but with less overt language than in decades past. You can see this with the Front Nationale in France and the Alternative für Deutschland (AfD) in Germany.

Now, what you described in your question (something ideologically different but shares similar foundations) is what's called post-fascism. Post-fascism is not a single, unified ideology, but it's more of a phenomenon. Many ideologies can be described as post-fascist ideologies, but they may not be primarily fascistic by nature. This is because the ideology may be merely inspired by elements of fascism, or draw upon certain aspects of fascist ideology. For example, Argentina is the country of my study. There, the ideology of Peronism is not fascist, but it certainly is post-fascist. It draws from many sources, fascism among them. Where Peronism breaks from historical fascism and neo-fascism is that violence no longer has a central role. Violence both as a means to an end and the end itself was absolutely vital in European fascisms. Yet in Peronism, at least in its original form in the 1940s and 1950s, violence does not serve the cathartic, regenerative, at times almost spiritual purposes that it does in German and Italian fascisms, making Peronism distinct from these earlier forms. Now, some Peronist groups by the 1960s had reintroduced violence as serving a major role, moving these groups closer to neo-fascist groups.

For further reading, I would definitely recommend Robert Paxton's Anatomy of Fascism. Furthermore, anything by Federico Finchelstein would be good reading as well. Previously, he's focused a lot on his home country of Argentina's history with fascism, as well as connections between Argentina and Italy. But more recently, he's written more about fascism in a general sense. A Brief History of Fascist Lies and From Fascism to Populism in History would be good.