What is the difference between a 'republic' and a 'commonwealth'?

Answer 1

It is a matter of interpretation, if they have both been admitted to the Union by law then the name is not relevant. They, what ever they might choose to refer to themselves, are equals and are considered as States of the Union. The use of the term by Puerto Rico is unique and no matter what they may choose to call themselves they are by law not a State, not a Commonwealth or any fancy high falluting title like Estado Libre Associado. The US Supreme Court rejects the term and continues to call it by the term assigned by International Law which is an inincorporated Territory. To move beyond or out of that status requires a referendum, the approval of the US Congress and the signature of the President of the United States.

Answer 2

Nothing. :] They are the same.