It's much easier to work the game in your head with descriptive notation, so it's good if you're reading the game. Descriptive notation lets you know overtly which piece took which, like PxQ. Having said that, it's not used much anymore, you'll only see it in old references.
Algebraic notation was adopted by FIDE because most of the world used it, so it was international, even if the letters used were different in different countries. It was pushed in the US in the 70s with Fisher's success, since chess was booming then, and so were book sales. The book publishers didn't want to have to print two different versions of the book for different notation markets. It really caught on in the US once chess-playing computers and software came out, since most of those used it.
Algebraic notation's first recorded use was in The Noble Game of Chess by Philipp Stamma in 1745: