Authority is the legal right for people to give instructions and commands to people who are supposed to follow their authority. They can give orders, and those who are under the authoritative figure are expected to follow. Responsibility is something that those in authority have. They are supposed to make sure that things will go according to what they have planned.
For example, it is their responsibility to make sure that the company will be able to reach the quota for that month. The more authority you have, the more trust you will also get. If something goes wrong, the person in authority will be held accountable for what occurred.