Nazism certainly is a form of fascism. However, Nazism had aspects that are not necessarily part of fascism. One way to put it is to say that Nazism was a form of fascism that emphasized racism and anti-Semitism.
Fascism is an ideology that emphasizes that people's first loyalty must be to the state and the leader, not to the individual. It is an ideology under which all people must see themselves as essentially similar. This is where the term "fascism" comes from. It comes from the idea of "fasces," which were bundles of sticks tied together with an axehead protruding from among them. They symbolized how things that are weak on their own (the sticks) can be very strong when united with others like them. In a fascist system, the individual people must band together unquestioningly and must make themselves identical with one another. In this similarity and commonality of purpose comes power.
As you can see, this does not necessarily assume racism. People can feel that they are identical and can owe loyalty to a state even if they are of different races. The same goes for religion. In Nazism, however, the idea was that people could not be identical unless they were racially identical. This led to a very virulent form of racism and anti-Semitism.
Nazism is a form of fascism because of its emphasis on the idea that people should be identical and should be completely loyal to the state. However, it is different from other forms of fascism because it believes that people of different races and religions cannot be identical enough to form the metaphorical fasces.