To confuse fascism and Nazism is perhaps understandable -- both were dictatorial, antidemocratic movements. But there are important differences between them as well.
First, let us unravel the terminology.
Revolutionary groups and whipping canes
Fascism originated in Italy, under the leadership of Benito Mussolini. Its name -- not its ideology -- is in part derived from certain revolutionary workers groups in Italy at the end of the 19th century, fasci revoluzionari (= revolutionary groups). When Mussolini started his movement in 1919, he called his first fascist groups fasci di combattimento (= combat groups).
As it happens, the similar-sounding Latin word fasces (signifying a bunch of whipping canes, bound around an execution axe) was an established Roman symbol of power. In early Rome, the entire contraption used to be displayed ritually before the powerful Roman consuls, who before 300 BC had the sole power to pass judgement as well as to mete out punishment (whipping by cane or beheading by axe). After 300 BC this combined judiciary and executive power came to an end, when Roman justice was reformed. But this didn’t bother Mussolini, who used pictures of fasces as his own symbols of dictatorial power.
Hence the word fascism has two "separate but equal" roots: the Italian word fasci and the Roman (Latin) fasces.
National socialism
Nazism is a contraction of the German word Nationalsozialismus (derived from the official German name of Hitler’s party, Nationalsozialistische Deutsche Arbeiterpartei, NSDAP). It stands for the totalitarian and racist pseudo-ideology under which the Adolf Hitler’s German Third Reich was ruthlessly governed.
During the 1930’s, political analysts in the democratic West were horrified by both of these antidemocratic creeds, Italian fascism as well as German Nazism. But they took care to keep them conceptually apart.
However, this was not the case in Stalin’s Soviet Union, because the full name of the Soviet Union has one word element in common with Nazism (Nationalsozialismus), the word socialism -- USSR is read out as the Union of Soviet Socialist Republics. Stalin, who saw himself as communist and socialist (and who in other respects was just as murderous and totalitarian as Hitler), was opposed to the competing totalitarian ideology of Nazism (= Nationalsozialismus). But he didn’t want to use the word in his anti-Nazi propaganda, because it contained a "good" element -- "socialism".
Hitler is a damned Nazi! -- Hell no, Hitler is a damned fascist!
So from the 1930's onwards a curious situation arose: when the West lashed out against fascists, they meant Mussolini’s Italians, but when the Soviets expressed anger against the fascists, they meant Hitler’s Germans. Communists in other countries followed the Soviet-established political vocabulary, using the word "fascist" when they actually meant "Nazi".
This difference in political terminology remained in place even after the war. The West celebrated its victory over Nazism, while parades were held in Moscow in honour of the glorious Red Army that had vanquished fascism. Both just meant the same thing -- that they were happy to be rid of Hitler.
An ideological litmus paper
When discussing historical WWII events during the post-war years, you could easily tell if the speaker or writer was inspired by communism. If he or she talked about Nazis as “fascists”, then the argument or point of view had in all probability originated in Moscow.
Unraveling the fuzz
Both fascism and Nazism are founded on fuzzy ideas, but this does not justify confusing them or treating them as identical.
Mussolini’s fascism had a shade closer resemblance to some kind of crackpot political ideology than Hitler’s Nazism, which was not based on much else than blind racial hatred, efficient militarism, and ruthless application of totalitarian power in the interest of the Master Race.
Corporatism –- medieval guilds warmed over
The Italian fascists regarded both parliamentary democracy and socialist class struggle as elements that were bound to cause divisiveness in a nation. Hence they introduced the idea of corporatism, a kind of modernised version of the medieval guild system. Here representatives of all trades and industries, employers as well as employees, could settle matters based on mutual understanding. Of course, in reality this was mostly ideological window-dressing. Mussolini was Il Duce (= The Leader) and had the last word.
In general, fascism was an appreciably lighter version of a dictatorial anti-democratic system than the mercilessly brutal Nazism. Fascist Italy never became completely totalitarian, nor did it commit mass murder on the scale of the Nazis’. The monarchy was intact and the bureaucracy, the military and the church remained as complementary power centres. Originally there was no racism in Italian fascism. Due to Hitler’s influence this unfortunately changed toward the end of Mussolini’s regime.
An inappropriate euphemism
The ideas of Italian fascism popped up among freakish movements in several European countries during the 20’s and 30’s, often vanishing after the war. They were applied during Franco’s and Salazar’s dictatorships on the Iberian Peninsula, where they managed to survive WWII by three decades, curiously enough. But Hitler's Nazis were far too cocky for ever terming themselves fascist. To the Nazis the Italian fascists were soft and ineffectual sissies, whom Hitler constantly had to save from various troubles that they always managed to get themselves into.
Is there any point in differentiating between these two historical evils? I strongly think that there is. Whatever Stalin and the communists may have said in the past, it is in my opinion hardly fair to the victims of Nazism to smooth over their horrors by euphemistically renaming their vicious Nazi murderers, making them look like mere 'fascists'.