Expanded from an earlier answer of mine

There were major differences between the ideology of Stalinism and National Socialism.

The idea that Stalin and Hitler represented two sides of the same authoritarian coin was one that dated back to when the actual dictatorships existed. The German Marxist emigre Otto Rühle would maintain in works like "The Struggle Against Fascism Begins with the Struggle Against Bolshevism" that Stalin's state was simply "Red Fascism" while Hannah Arendt would argue that both dictatorships exerted power in near-identical ways by fragmenting and terrorizing populations. Much of this critique actually picked up on the fascist concept of a totalitarian order in which the state organized and regimented society. While some of Mussolini's ideologues use totalitarianism in a positive sense, the term acquired a pejorative sense by the 1940s. The totalitarian theory emerged as one of the dominant paradigms in US academia during the first half of the Cold War and it enjoyed a wide currency among intellectuals. Counter-intuitively, totalitarianism held sway among a broad political spectrum, ranging from far-right anticommunists like the John Birchers, to centrist Cold War liberals, and disaffected Trotskyites and others on the Left like the Schachtmanites who felt that the Soviet experiment went off the rails into dictatorship because of the ideological deficiencies of the helmsmen.

The problem with totalitarianism is these comparisons only work on the most superficial levels. Both regimes had leadership cults, single party systems, secret police, and other benchmarks of the modern dictatorship for sure, but how each of these served as a mechanism or authoritarian rule varied quite widely. The NSDAP for example, sought to coordinate and bring in a broad swath of German society and co-opt existing structures. The Gestapo and other security organs recruited among various existing German police and disaffected professionals (Wildt's "uncompromising generation" of radicalized young men) while the Cheka/NKVD created its own police organization more or less from scratch. The trajectory of the CPSU under Stalin operated quite differently with it creating a number of structures rather than coexisting with existing ones. Stalin's leadership cult was a different matter than that of Hitler's. A cult of personality in an avowedly Marxist-Leninist state was itself a contradiction; Marxism was about the evolution of broad social forces, not the individual leader transcending them. Stalin instead preferred his depictions to not be Stalin the man, but rather Stalin as the symbol of the cause. In contrast, the Hitler cult was deeply personalist and wrapped up Hitler's own self-inflated biography as a genius with the a political philosophy of Führerprinzip. Hitler himself was an load-bearing member of the National Socialist state as there was very little thought given to succession and none of his entourage had close to a mass following to continue the cause. The Soviet dictatorship in contrast not only survived Stalin's death but reverted to the type of collective leadership with a "first among equals" like Khrushchev that had characterized the Lenin period.

Racism and other forms of ethnic-based discrimination were another departure between the two ideologies. While there was a low-grade antisemitism within the Soviet system as well as an increasingly patronizing approach to non-European minorities within the USSR, biological racism did not have much of an official sanction within the USSR. Quite the opposite, prior to 1941, open outbursts of antisemitism were a disciplinary offense and the Stalinist state associated racism as a tool of fascism. Stalinist repression of Jews may have been encouraged by lingering antisemitism among the state authorities, but there needed to be some other rationale to officially justify it. The postwar antisemitic turn like the Doctor's Plot was justified not because Jews were innately traitorous, but because the state alleged a Zionist plot was forming within the USSR to undermine the state. This is quite different than the Third Reich which placed biological racism at the center of many of its various policies. The purging of Jews from German life was not because of the actions of a few, but because of who the Jews were in the Nazi racial schema. In the USSR, a Sovietized Jew (that is non-practicing and holding the "right" kind of politics) could achieve a degree success within the system, most famously in the example of Molotov, even if antisemitism made Jewish background a dangerous liability. The idea of a Nazified Jew was antithetical to the whole National Socialist Staatsidee. Even those individuals with partial Jewish heritage like Erhard Milch had to both demonstrate their indispensability as well as have the right social networks to avoid summary dismissal and discrimination.

Nationalism also shows another point of divergence. It is not too controversial of a point to claim that the Third Reich was a hypernationalist state and Third Reich ideologues often claimed all of the German heritage as precursors to the movement. The Stalinist approach to Russian history and nationalism was much more selective and only gathered steam during the Second World War. Only a few tsarist figures were reappropriated as Bolsheviks born in the wrong era; typically these were military generals, intellectuals, and artists. And although the Great Russian nation became the first among brothers within the Soviet family of nations, the Soviet state did not try to erase recognized nations either. The policy of korenizatsiia (indiginzation- termed by Terry Martin Soviet affirmative action) allowed Soviet national minorities a degree of power within their own SSRs. Granted such power was mediated through the Communist Party and the local state bureaucracy and it did presuppose a degree of supplication to Soviet power, but it was still a mechanism for non-Russian participation in the Soviet experiment. The Nazi approach to national minorities had no such structures aside from extemporized categories of racial lieutenants like the Slovaks. Additionally, while Stalin's ethnic cleansings such as that of the Crimean Tartars could be quite severe, the USSR never made the conceptual leap that these ethnic groups did not deserve a right to exist. Nazi thinking of defined racial enemies was genocidal in its impulses from the very start and the right conditions made an already violent form of discrimination truly murderous.

The realm of economics was also a major point of departure despite the S in the NSDAP. There was an anticapitalist wing of the NSDAP, especially in the early days of the movement and centered around the Strasser brothers. But anticapitalism is not the same as Soviet Marxist-Leninism, especially in the German context. There was a strong strain of German nationalist thought that was hostile to American or British-style mammonism and consumerism. The German university system for example, incubated a number of intellectuals who argued that capital needed to be aligned to the nation and the state. The hyperinflation of the early 1920s as well as the advent of American consumerism added a further reactionary critique of capitalism. The NSDAP picked up the intellectual crumbs of this thought and things like the NSDAP party platform were a reflection of the increasingly marginal importance of the anticapitalist wing. The Bamberg platform was less important than the political maneuvering behind the scenes in which Hitler brought the Strassers to heel.

Once in power, Hitler pretty much abandoned the anticapitalism of the party's earlier years. Outside of Aryanizing Jewish property and a half-hearted campaign against department stores, the Third Reich did not do that much to alter capital relations in the country. The Nazis generally respected Aryan private property and entrenched German industry's preferences for cartelization and a close relationship between industry and state. The state would occasionally work against the interests of capital, but in most cases this was related to rearmament's needs, not some deep-seated ideological hatred of capitalism. The contradictory and half-formed ideas of National Socialism on the economy were that the type of dog-eat-dog competition was a feature of capitalism created by Jews and their allies. National Socialist ideologues posited that a racially-reformed German state would purify capitalism of its vices and create a harmonious Volksgemeinschaft. The idea of reforming capitalism was anathema to the USSR throughout its existence. Even the introduction of minimal or limited markets of the NEP period and Gorbachev era were not welcomed by Moscow and existed i an uneasy tension with the state.

Other areas also show broad differences between the two states. Soviet developmentalism and industrialization under Stalin reflected various precepts present in Marxist-Leninism with an aim to a certain end, namely the erection of socialism and the creation of a communist utopia. Massive industrial projects like Magnitogorsk were a Promethean attempt to bring about an industrial transformation of the country. The mass projects of Hitler served to glorify the existing order, even if they had an ideological gloss of being in the service of a "thousand year Reich." Moreover, the reams of social history produced on Germany since the 1960s and the USSR since the late 1980s has shown that the lived experience of both dictatorships was markedly different.

This is quite a question, you should probably try asking it in r/politicalscience too, but i will try to answer it to the best of my abilities.

I can try and lay out the core ideological tenets of the two ideologies and draw the differences.

First of communism :

While communism has a few different offshoots : marxism-leninism,orthodox marxism, maoism, stalinism and various others, i will assume your focus was probably on the soviet type so i will explain, in short, marxism and marxism-leninism.

The philosophical ideology that all of communist ideology is based on is dialectical materialism : the idea that thought arises from being and not the other way around ( i am therefore i think instead of Descarte's I think therefore i am). This means that human thoughts only have a meaning as they are interpreted trough actions and the way those actions effect the world in the social,political,economical and historical context. Dialectical view means seeing the world as a system of opposing structures that either complement or are in conflict with each other (prey and predator,body and mind, rich and poor, master and slave etc.).

Marxism is the application of dialectical materialism to human society, resulting in the antithesis between the ruling class and the servant class, between the oppressors and the oppressed. According to Marx, this antithesis existed throughout human history ever since the agricultural revolution (in ancient times there was masters and slaves, during feudalism you had nobles and peasants and in modern times you have the bourgeoisie and the proletariat (the ones owning capital and the ones,without capital, that exchange their labour for subsistence from the capitalists). So, in marx's ideal world, the proletariat would collectively own the means of production (capital) and the oppressing class would disappear entirely thus starting a new era of a classless,moneyless,stateless equal society where everybody "would give according to his ability and would get according to his need". In order to get to this "upper" communist state as Marx called it, society has to go trough a "lower" communist phase that Lenin called the socialist state : an intermediary phase where the state would still exist and the proletariat, after overthrowing the ruling bourgeois class, would become the oppressors of the once ruling class and would use the tools of the "old state" in order to change society into one more conforming to communist ideas and make the preparations for a stateless communist society. Marx didn't go into a lot of specifics about this transitional state and this is where most of the divergence between the different communist doctrines come from. Lenin believed that the transitional state would be one where the representatives of the proletariat would hold supreme power and use the tools of the "old state apparatus" to change society up until an unknown point in the future where there would be no longer a need for government coercion and the state would "wither away" (this of course, never happened in any socialist country) and this is where the one party system that many socialist countries inspired by the USSR came from.

Now, onto fascism. Fascism was a movement that stemmed out of 19th century nationalism and quite a few philosophical ideologies. During the 19th century, existentialism gave birth to new lines of thought that put into question the ideas of the Enlightenment : critical thinking,tolerance,secularism,equality,individualism etc.. This antirationalist movement put into question ideas like objective truth,human equality,democracy,liberalism and morality. Coupled with the surge of nationalism as most peoples around Europe forged a national identity under the rule of the various empires this lead to the first fascist ideas. While someone more knowledgeable than me could shed more light on the evolution of these ideas from here on out i can present the end product: After the first world war and the economic turmoil that followed it, many people, especially in Germany,Italy, Romania and Hungary were drawn to the antirationalist elements of fascist ideology as they started to question the democratic institutions and western liberalism as, in their eyes (and somewhat justified considering the harsh times these countries where in) democracy and liberalism failed and fascism was the alternative. Now onto the ideology itself: Fascism (word that comes from the italian word for "bundle of sticks") is an ultra-nationalist movement that put the nation above everything else and taught that a citizen's most important duty in life is to serve its country. It emphasized the importance of national values (hence the difference from country to country) and viewed minorities as a threat to national hegemony.

It mostly rejected christian values as they,in their eyes, lead to liberalism (while fascist dictatorships did maintain relations with christian authorities it was either a political move or to garner support from the people as a lot of people were still highly religious and the regular citizen wasn't exactly a fanatic ideologue) and put a lot of value on antic pre-christian beliefs because of their heroic myths where one person is presented as doing brave deeds for the good of their people (think greek,roman,germanic heroes etc). This would be the ideal fascist man : a brave warrior willing to sacrifice himself for the good of the nation. Fascists put a lot of value on self-sacrifice and the myth of the hero of the people that is there to lead the nation, hence the dictatorship and the cult of personality around dictators. Because of this warrior cult, fascist societies were highly militarized and there was no higher honour than to sacrifice yourself for the good of the country (again, something taken from ancient civilizations, especially Rome that had a highly militarized society, after all Mussollini's highest ambition was to recreate the Roman Empire). The hatred of minorities came mostly from the ultra-nationalistic culture instilled by fascism, as anyone outside of the nation was an outsider and thus a threat to the well being of the nation.This hatred and paranoia only amplified existing stereotypes and was backed by pseudo-science supported by the regime and put out by fascist thinkers.

Now onto similarities and differences : First of let's start with the ideal communist and fascist man or member of society : Both ideologies required a reformed society made from members educated in the respective doctrine : Like i said earlier, the ideal fascist man was a brave warrior willing to sacrifice himself for the good of the nation and the people rejecting liberal ideas and individualism, while the ideal communist man was a man who put the good of the community or the group above himself and rejected ideas like individualism and competition. While both are required to think of the collective first and of themselves second, they differ in the way that their respective societies value different traits (bravery vs solidarity, self-sacrifice vs compassion, militarism vs cooperation etc). A clear similarity between the two is the inherent totalitarian nature of these ideologies (while a branch of communism that beliefs the change from capitalism to communism can be made without a transition state exists, i believe you wanted to compare fascist and socialist states) : both require the concentration of power in the hands of one man(the commander of the people: "il duce", "fuhrer", "capitan","conducator" etc) or a group of people (the party) because of the drastic changes to society that both want to enact.

A huge ideological difference though, is the way they see people : Fascists definitely believe in the superiority of one nation over another as they believe that each nation of people has some inherent characteristics that makes them superior or inferior while communists see the nation-state as an artificial construct that only stands in the way of the unification of the oppressed and that the only real distinction between people is their socio-economic class.

As an end to this wall of text i will say that, while fascist and socialist states may seem similar at a first glance because of their totalitarian nature ( as totalitarian regimes usually employ similar tactics and most of the time are hijacked by power hungry individuals that have as their only goal to remain in power and not to enact an ideology) their world views are inherently different and this ideological divide was one of the reasons that they came into conflict during WW2.

If you have any questions i will be happy to answer them