What are the advantages of using a process manager inside a Docker container?

That's the whole idea, to use systemd (or upstart etc) inside a container. Docker is used most of the time to run a single service per container, so for a complex site you would need a container for the web server and another one for the database server. With a process manager you could run both in the same container.

Which approach is better is opinion-based. Though from what I've seen running systemd inside Docker isn't easy as of June 2014, for example there's bug #3629 - "running systemd inside docker arch container hangs or segfaults". For more details also read "Running systemd within a Docker Container".