When I worked with docker containers, I noticed that they seem to present a full OS, with a FS that you had to manage.

I had to change /etc config files, mange where logfiles went and how they were handled, all of this is described on a diffing filesystem.

I might be misinterpreting what you mean though. I have no experience with OpenVZ and little with Docker.

This was mentioned many times, but still: this is about "best practices" or common use case. OpenVZ or Docker (or even full VMs) can interchangeably be used to solve a given use case