The term fascism comes from the latin word fasces, which is a bound bundle of wooden sticks that was a symbol of the Ancient Rome society which Mussolini aimed to recreate.
Nazism, however, is short for national socialism, so it appears (to me, at least) to have a broader meaning.
So, why is now fascism a wider term than nazism?
Since you mention the origin of the word "fascism", I think you may find this answer of mine of some interest.
As for why we use "fascism" to describe national socialism, we should not forget that fascism appered as a movement in 1919 and become a noticeable political force during 1920-21. Mussolini was appointed Prime Minister in 1922 and progressively established a one-party regime during the 1920s.
Which means that, broadly speaking, the Fascist Regime was established politically and in the public mind at least a few years before the others examples of "fascism". For this reason, the other regimes, or political groups which attempted to pursue a similar direction (or were perceived in such way - for a review of the various "fascist" or "para-fascist" groups see R. Paxton's Anatomy of Fascism or Payne's Fascism) fell under the same denomination.
It was certainly reasonable to group those organizations, political forces and, loosely speaking, ideological patterns, under the name of their first and most successful (until the late 1930s at least - and I obviously mean successful from the fascist point of view) incarnation.
The collocation of many of those forces on the same side during the Spanish Civil War and then during WW2 contributed further to their identification with "fascism" as a general phenomenon.
Truth be told, there is nonetheless a significant difference between the use of "fascism" in a scholarly context and the public use of the term. Italian Fascism and German National Socialism do share a few points of similarity which go beyond the surface of parades, flags, leadersim, and collective social militarization. But one doesn't generally use "fascism" for any "right wing dictatorship". Especially in Italian scholarship, the term is generally reserved to the two "accomplished" regimes and called into question already for Spain - usually applied only with major reservations to authoritiarian regimes such as Horty's, Antonescu's or Salazar's, which were all arguably right-wing.
In this sense, one should remember that the attribution of a "fascist" brand to a regime is not really an unappealable judgment - the term, with what it designs, is an instrument to understand the workings and characters of a political system, and it can be applied to other examples outside of the canonical ones. But in general, such attribution is done under the frame of an analysis rather than as a definition.
Last, if you are interested in the historiography of the use of "fascism" as a political denomination during the second half of XX Century - of which I know close to nothing - I would nonetheless consider examining the issue from a different perspective, since the common use of the terms "fascist" and "nazi" doesn't in general follow from historiographical concerns, nor from considerations on the actual origin and general value of the original denominators.