The legal ramifications of firing someone for being a member of the Nazi party, versus being a member of BLM, are very different. The reason is that (to my knowledge) Nazi is not a protected class. Now, being a member of a protected class does not mean you can’t be fired. It means you can’t be fired for being in that class. An employer who fires someone for being in BLM would find themselves in court arguing that they fired the person because he was in BLM, and not because he was black, versus a skilled lawyer who would be working very hard to convince the jury otherwise. A good lawyer would have a field day, and the employer would be quite vulnerable. By contrast, there is no protected class status for being a Nazi, so there is no comparable legal issue.
There’s also the obvious fact that BLM, aside from a minority (arguable in size) of its supporters, is not inherently violent, racist, and supremacist in nature.
That could be debated. Protected class cannot.