The "different country" factor is far from an irrelevant one indeed, especially for two political formations that rose in the context of early XX Century nationalism - that is in an environment where national characters, either cultural, religious or racial were enhanced in the form of political identifiers.

But, broadly speaking, even among liberal or democratic forms of government, one would accept that there are significant elements of distinction between different countries (think for instance of the UK and the US).

One of the reason why most observers agree that National Socialism and Italian Fascism share enough similarities to warrant the combined definition of "fascist regimes" is exactly the fact that many of their differences can be explained with the relative "national character". In both Germany and Italy (the latter is the one I know better - I hope any possible mistakes about Germany would be corrected) there was an effort to re-create some form of national unity, around some fundamental core of values. The National Socialists found their original point within a series of pre-exisiting theories, those grouped around the idea of volk: the people, meant as a racial, cultural and natural community, with its own religion and consequent political formation. For the volkisch movements, the nation was first and foremost an expression of a common heritage - the political form was subordinate to the racial and cultural expression. For this reason among others, the National Socialist movement paid far less attention than Italian Fascism to the formation and organization of a new state, as for the Nazis the state was reduced to an instrument in the affirmation of the racial-cultural-national community.

For Fascism, which had inherited this concept from the Nationalist formulation (mostly due to jurist Alfredo Rocco), the State itself was the concrete form of the national community - a community that was more culture-organization than ethnical-racial (even if certain racial components became more prominent in the late 1930s after the Ethiopian War). According to Rocco, a proponent of the juridical doctrine of the state, the state (the ultimate organized community) was the source of any right (public and private) and individual rights were only "a concession made by the state for its own interest" - in the same way a person wearing shoes does it for their own good, not because of the individual rights of the feet. As you can see, the juridical state is alternative to jusnaturalism.

Therefore, while for National Socialism the formulation of a state doctrine remained fairly limited and even juridical doctrines revolved around the idea of a natural community that found its expression within the persona of the leader, for Italian Fascism, the state doctrine became its most "advanced" doctrinary formulation (comparatively speaking).

The State-Nation was never an instrument, but the ultimate end, the true expression of an organized society and the leader, the Duce, rather than the direct expression of a racial-cultural-national community, was the incarnation of the hyerarchic principle that inspired the organization of the state.

From this difference you can see how Italian Fascism would place much less attention to racial issues. But you should not anyways overestimate the intentional ideological differences. Both Fascism and National Socialism were remarkably shallow from an ideological point of view; and while the Nazis experienced a far more rapid and more extreme radicalization, many of their features remained a result of their original environment (the radical right movements during the war had evolved in different manners in the two countries - for instance antisemitism was very much the exception even among the radical right in Italy, while it was common, if not commonplace in Germany) and of their organic growth.

On this ground, many have explained the slower and incomplete radicalization of Italian Fascism with the persistent of competing centers of power within the country, that either resisted or slowed down the expansion of the Regime: the King, the Military and the Church continued to play a significant role in Italy during the whole political experience of Fascism.

On the other hand in Germany the rise of the National Socialists conincided with a progressive vacuum of power: first the Empire followed by the weaker Weimar Governments, than the death of the President, the fire of the Reichstag, the almost immediate concession of special powers to Hitler (for instance when Mussolini asked a decree for the dissolution of the Chamber on the day of his appointment to Prime Minister, the King outright refused him by stating that he intended for Mussolini "to rule with the Parliament").

As you can see, while certain traits can be described in ideological terms as "abstract characters" of the two movements, many others decisive features were the result of their actual evolution - as Angelo Tasca wrote already in the 1930s "to understand Fascism is to tell its history". By which one means that an ideological explanation of the two movements is destined to remain incomplete unless their actual historical process is subject to analyisis. And for similar and more detailed considerations, see also R. Paxton's Anatomy of Fascism.

As usual for the few bits on National Socialism, I used Kershaw's Hitler and Evans' The Third Reich - for Fascism mostly, De Felice's Mussolini; Gentile's Le origini dell'Ideologia Fascista and Il mito dello stato nuovo