Here is the basic difference between run and start command.
Run command is used to create a new container of an image, and execute the container. You can create N clones of the same image. The command is: docker run IMAGE_ID and not docker run CONTAINER_ID
Start command It is used to Launch a container previously stopped. Such as, if you had stopped a database with the command docker stop CONTAINER_ID, you can relaunch the same container with the command docker start CONTAINER_ID, and the data and settings will be the same.
I hope it would resolve you query.