首都 equal to capital. And Chinese using '首都' to describe the capital of other countries .Like 'the capital of America' is '美国的首都' ,not '美国的京城'. The word of '京城' is historical('京' was described as the place of the emperor(also called Tianzi in ancient time) located) and Chinese only use '京城' to describe our own capital. Wish this anwser will help you.

The first one means "imperial city", while the other means "capital". However, the latter is now used. We don't call London by the first term since the monarchy is no longer absolute. Same reasoning for Tokyo. However, before WWII, it can be used to described in such a way.

首都 is much more popular in modern Chinese while 京城 or 京师 refers to capital also but it may be used in ancient time more often. Interestingly, the Portuguese name of 葡京大酒店（which lies in Macau） is "Hotel Lisboa", here 葡京 refers to 葡萄牙的京城, which equals Lisboa（里斯本）. Maybe Chinese people were used to call the capital of other countries as 京城 when we started contacting with the world.

口语中多用“首都”，“京城”是一个偏方言的名称，只需要记住“京城”就是指的“首都”，两者意思一样。but，最重要的是，所有用“京城”这个词的地方都可以替换成“首都”，但是，所有用“首都”这个词的地方，就不一定可以用“京城”来替换了。

“京”指一国之都，在不同的历史时期中国有着不同的京城，当然现在的“京城”就是我们的首都了，只不过更口语打趣