on-failure will issue a restart if the exit code indicated a failure, whereas unless-stopped behaves like always and will keep an instance running unless the container is stopped.
You can try with the hello-world to see the difference.
docker run --restart on-failure hello-world will run once and exit successfully, and running a subsequent docker ps will indicate no currently running instances of the container.
However, docker run --restart unless-stopped hello-world will restart the container even if it successfully exits, so subsequently running docker ps will show you a restarting instance until you stop the container.
CONTAINER ID IMAGE COMMAND CREATED STATUS PORTS NAMES
4d498ebd13a6 hello-world "/hello" 2 seconds ago Restarting (0) Less than a second ago modest_keldysh