TL;DR: Marxism and Nazism are just two competing ideologies, so they thought of some dogmatic ideas to criticize each other. Their goal was to win the competition between the two ideologies, so virtually none argument was based on any solid, scientific research; instead, most (all?) of the arguments appeal to the senses, not to the minds.
In order to completely answer this question, it may require a book to cover all aspects, so please excuse me for losing accuracy in favor of compact answer.
First off, Fascism and National Socialism are different things: Fascism is a regime (type of government), while National Socialism (Nazism) is an ideology. Between WWI and WWII they used to be tightly related, but they are not the same.
Note: Regardless "National Socialism" contains a word "Socialism", which suggests a counter-intuitive understanding, it has nothing to do about the type of the government:
Our adopted term 'Socialist' has nothing to do with Marxist Socialism. Marxism is anti-property; true Socialism is not. — Hitler, 1930 (ref. Carsten „The Rise of Fascism“, 1982)
The Kommunist counterparts are Socialism (a regime) and Marxism (an ideology). They also used to co-exist in the Russia and the countries it occupied.
As far as I understood your question, we are not talking about the social system, but about the ideologies.
Nazism is an ideology of racial supremacy. It included several aspects like "racial hierarchy" where Aryans supposed to be the "master race", while others are "lower" ones. The special attention was directed towards antisemitism as Jews were declared the "worst" race.
Marxism is an ideology of class supremacy. It included several aspects like "class hierarchy" where Proletariat supposed to be the "master class", while others are "lower" ones. The special attention was directed towards Bourgeoisie as they were declared the enemy who exploits the proletariat.
Being two competing ideologies, both Nazism and Marxism have elaborated some dogmatic ideas to confront each other. Here can be a huge list of arguments, but they are really unimportant since all of them are artificial, deliberately created in order to withstand each other.
Anyway, the ideology struggle had not in any way prevented the Soviet Russia and Nazi Germany to be loyal allies in the period of 1933-1941.
An important aspect is that the vast majority of Russian Intelligentsia of mid-1910's who led the October revolution were ethnic Jews. This may make someone think that Marxism is a kind of Jewish conspiracy. However, Marx's own views were arguably anti-Semitic:
What is the worldly religion of the Jew? Huckstering. What is his world god? Money. — Marx, „On the Jewish Question“, 1844; (ref. Francis Wheen — „Karl Marx: A Life“, 2001)
Yet another important aspect is that after defeat of Nazi Germany, the Soviet lawyers in Nuremberg trials have insisted to add Anticommunism in a list of Defining
Characteristics Of Fascism.