Good question! I needed to research a bit.
They essentially mean the same thing and the difference comes mostly in usage. Vacant is defined, in part, as empty.
In use, empty is mostly used with objects. "The trash bin is empty", "The envelope is empty" "My stomach is empty"
Vacant refers more to a 'sense of being'. "The parking lot is vacant", lacking cars to serve it's purpose. "His eyes were vacant", to mean that no 'life or soul' could be seen in his eyes. That seat is vacant, you are free to sit there". "I have one vacant apartment, if you wish to view it".
However they cross. An apartment can be empty, to mean that no one is there. But, an apartment can also be vacant, to mean lacking tenants or fulfillment. "My life is empty" is using empty as you might use vacant. Lacking fulfillment.
I hope I've helped here,