Petter, it is faster for me to type in English. 毕竟 means something follows logically from another thought. In English, you'd say "after all." For example, you can say "His English is really good. He is, after all, American." (他的英文非常好。他毕竟是美国人。) 究竟, however, is an emphasis term when you want to know the answer to a question - it's like saying "what in the world happened here?" 这里究竟发生了什麽事？

說白了 畢竟：就是 究竟（書面語）：到底（口語）

我以上写的如果有语言方面的问题的话，请修改，谢谢