Beijing = Northern Chinese pronunciation

Peking = Southern Chinese pronunciation

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.)