But neither one is something we can pronounce correctly anyway, I suppose.
Well then, Florence or Firenze?
Greece or Ελλάδα;
Japan or 日本国
Beijing and Peking are different romanizatiins of 北京
Beijing is Pinyin and Peking is Wade-giles
Both are pronounced the same way
Well, you're one for three. Beijing and Peking are different romanizations of 北京. It is not true that beijing is pinyin and peking is Wade-Giles, nor is it true that they are pronounced the same way.
Wade-Giles would be pei-ching. (Pinyin is indeed beijing.)