As we could see from the different posts, it is nearly impossible to compare those two parties. The US-party exists much longer and has an important tradition, whereas in Germany the Republikaner are a pretty new party. Second of all, you have to consider the size: In the US, the party is one of the two major parties and you could even consider it as a personal political attitude. In Germany the party is often too small to even get some seats in the Bundestag or Landtag.
The third point is, that you can not compare the two parties, because in the US you vote for a candidate and in Germany you vote for the party and its policies.
I think the political culture plays an important role as well, because in Germany it is not too welcome to be a bit more extreme than the others are and especially not when it goes into the direction of being patriotic or even nationalist. All in all we figured out that "being republican" means something totally different in the two countries.
pictures from democraticunderground.com and politik-visuell.de
