They are the same place. Here an abridged Wiki entry:
Peking is the traditional English name. The term originated with French missionaries four hundred years ago and corresponds to an older pronunciation predating a subsequent sound change in Mandarin.
In China, the city has had many names. Between 1928 and 1949, it was known as Peiping, literally "Northern Peace". The name was changed � with the removal of the element meaning "capital" (jing or king) � to reflect the fact that, with the Kuomintang government having established its capital in Nanjing, Peking was no longer the capital of China, and that the warlord government based in Peking was not legitimate.
The Communist Party of China reverted the name to Beijing (literally means "Northern capital") in 1949 again in part to emphasize that Beijing had returned to its role as China's capital.