There are times when these mean the same thing.
At other times, vacant means "unused" all of the time and empty means "unused at this moment".
The glass is empty. (I'm not using it now.)
The house is vacant. (No one is using it / living it in permanently.)
The theater is empty. (All of the people went home.)
The theater is vacant. (They don't use it for performances /movies any more. It's an empty, unused building.)
The empty (parking) lot. (All of the cars are gone now. But they'll be back tomorrow.)
The vacant lot. (No one uses it for anything anymore.)
An apartment or motel can have vacancies: rooms available for people to move into.
The apartment can be vacant (no one is using it), but it might not be empty (it still has the furniture from the previous owner in it.
I went away empty-handed. ( I did not get what I wanted.)
If someone is in shock, he might have a vacant stare. (He's not thinking about what he is seeing.)
Empty promises. There are promises, but they are meaningless / unfulfilled.