What's the difference between ]a,b[ and (a,b)? Is there any difference between $]a,b[$ and $(a,b)$? If there is no difference, what would be the motivation of using $]a,b[$ over $(a,b)$?

There is no difference. They both refer to "the open interval from $a$ to $b$." The "advantage" of using $]a,b[$ is that it can't be mistaken for an ordered pair. 
