TL;DR -- Nazis want to kill people, commies want universal peace and harmony and individual commies have been willing to kill people to achieve those goals.
Nazis refer to the Nationalsozialistische Deutsche Arbeiterpartei aka NSDAP aka National Socialist German Workers Party, a very specific group. Nazis are not the National Socialist party of Italy or the US. Showing Nazi symbols can be taken as support for the group and the various crimes they committed -- accessory after the fact.
Communism on the other hand is a political philosophy, and is not specific to any one group (for instance the USSR was against the Thailand communist).
Whatever their other goals or means used to achieve those other goals, we know that Nazis had goals we find inhuman and that they used means we consider criminal in their attempt to achieve those goals.
Communism may require means we consider criminal in order to have any hope of achieving their goals, but their stated goals are not themselves inhuman.
That means that it is perfectly possible to be a communist and unwilling to advocate killing anyone, but not possible to be a nazis without advocating killing some people.
This difference between the two makes making laws against nazis symbols easier than making laws against communist symbols.