Having authority gives you the right to enforce something i.e. the school Principal has is allowed to enforce the rules of the school. Having the right means that you have the legal or moral entitlement. i.e. the school Principal has the right to enforce the rules. I hope this clarifies it for you :-)

Two sentences. I have the right to do this. This means that someone or something has permitted you. I have the authority to do this. This means that you control the action, it does not mean that someone has allowed you to do it.