In short: Docker service is used mostly when you configured the master node with Docker swarm so that docker containers will run in a distributed environment and it can be easily managed.
Docker run: The docker run command first creates a writeable container layer over the specified image, and then starts it using the specified command.
That is, docker run is equivalent to the API /containers/create then /containers/(id)/start
source: https://docs.docker.com/engine/reference/commandline/run/#parent-command
Docker service: Docker service will be the image for
