What is the difference between docker run and docker container run

They are exactly the same.

Prior to docker 1.13 the docker run command was only available. The CLI commands were then refactored to have the form docker COMMAND SUBCOMMAND, wherein this case the COMMAND is container and the SUBCOMMAND is run. This was done to have a more intuitive grouping of commands since the number of commands at the time has grown substantially.

You can read more under CLI restructured.

docker run no, we aren't even hiding it, it's staying as a permanent alias.

The rest, not any time soon. Maybe in a year or two if we're good about converting all > the docs to the new form, and communicating the new canonical way of doing things.

So, they are exactly same, just format changed, see discusstion about this PR: https://github.com/moby/moby/pull/26025