That's not correct, actually. There were two brothers who inherited Aldi, and they did have a falling out over cigarettes, but they actually split the company in two - Aldi Nord (North) and Aldi Sud (South). As the names imply, they operate the Aldi stores in North and South Germany respectively.

In other countries, either Aldi Nord or Aldi Sud operates the Aldi stores, but they do not directly compete with each other. The exception is the US, where Aldi Sud operates the Aldi stores and Aldi Nord operates Trader Joe's (which the original owner of Aldi bought from Joe Coulombe in 1979).