The Wailing Wall is the Western Wall. Jews have always called it the Western Wall. Wailing Wall is a somewhat derogatory term, probably coined by 19th century tourists to describe the place because they saw Orthodox Jews mourning there for the destruction (some 1800 years earlier).
There's an old joke from the 1960's. An American tourist lands in Israel and wants to go straight to the Western Wall, but doesn't know what it's called in Hebrew. In exasperation, he tells the taxi driver to take him to Jerusalem to the place where everyone goes to cry. They drive up to Jerusalem and the taxi stops in front of a modern office building. The tourist says, no this isn't the place. The driver says, yes, this is where all the Jews cry - the income tax office.