There can be a few differences between Peking and Beijing. This can be just norm and my own feeling about this. I am not sure if there is an official explanation of this. But anyway…
Peking and Beijing are in the constant change for hundred of years. Beijing is the name we are using now though.
First in the 14th century, it was Peking. It means “peace in the north”. A very good meaning actually. Then in the 15th century, the capitol was moving from Nanjing to Beijing, and it was then Peking was changed to Beijing. It was to correspond with Nanjing, the former capital. Nanjing means “the capitol in the south” and Beijing is “the capitol in the north”. Peking was in use again in 1928 and change back to Beijing in 1949.
Listed here are the brief history and there are also something subtle.
Beijing is preferred now because when it was Peking in the early 20th century, China was under awful condition. War time and invasion back then.
A very good example can be this. An amazing TV series is called “All Quiet in Peking”. It is a story in the 1948 and 1949, right before the new China (PRC) was founded. We can tell directly it is a story from that period of time because it uses Peking instead of Beijing.
I know outside of the Chinese language, other language don’t always distinguish Peking and Beijing. But I do hope they do. Trivia things mean a lot, isn’t it?