Earlier quoted context omitted.
Docker and OpenVZ are not the same. Docker is single application focus. OpenVZ provides the entire OS in a container. OpenVZ has support for live migration.
OpenVZ is closer to KVM (full virtualization). OpenVZ container is a lightweight VM which shares host kernel, has persistent FS and traditional OS you have 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.