Both words mean people seek to settle disputes between two (or more) parties. And both refer to individuals who listen to the disputes. They go into arbitration or mediation because they can't agree.
Here's the difference: An arbitrator has power. Some agents have an item in their contracts with writers that says if they have a dispute they can't settle they will go into arbitration.
I like to say it this way: The word mediate, over the centuries, has lost its power. It has come to refer to a person who helps disputing parties come to a satisfactory conclusion. It's usually less formal than arbitration and is rarely binding.
I'm careful about the use of words.
The more careful I am, the better writer I become.