Both Rettich and Radieschen refer to radish.
Is there a difference? Is it a regional difference? Or different kind of raddish? Or is it just different names that refer to the same kind of generic radish?
This answer refers to everyday usage of the words, not necessary the biologically correct definition:
To me as a native German speaker from South-Western Germany, Rettich and Radieschen are two entirely different things (and I am not sure I ever thought the two could be biologically related until this question just mentioned them together and thus hinted at a relationship).
Rettich is a long, white thing that is large enough so your fingers just reach around it when holding it in your hand. Among other uses, it is eaten by some people in small spiral-shaped slices with salt. Other than that, it seems to appear in dishes not so originally from Germany such as kimchi and some types of sushi.
Radieschen are little, bite-sized pinkish-red balls that are white on the inside. They are often sliced and added to salad or eaten on top of buttered bread.
Radieschen are garten-rettich, garden rashishes, or in English, common radishes (Raphanus sativus), tart little red-skinned white spheres. Sometimes, it is called the German radish.
Rettiche is a broad category of plants, Brassicaceae, that includes common radishes (even the large daikon or "Oriental radish" cultivar), rape-seed, cabbage, and turnips.