A binding is a strip of fabric, usually cut on the bias, hence bias binding. It wraps over the edge of the fabric and shows on both sides.
A facing is only seen on one side of the fabric, usually the inside, and is an exact copy of the area it matches to.
However, there are ways to finish an edge that are a combination of both techniques. Sometimes called a narrow facing and is often found on bought clothing. It uses a bias strip to face the edge and can show on either the right side or inside of a garment.