It's almost the same. If you will co relate it to work, additional work is something you are required to do in order to perform your duties which is still within the scope of you contract. On the other hand, extra is something you are not required to do but you are being ask to, in order to perform a duty that may not be within your scope of contract. Therefore, additional is something expected while extra is something sudden in a way. I hope this helps a little.

They have exactly the same meaning. For example: "It's going to cost an additional $5." "It's going to cost an extra $5."