They are built around the same set of kernel features and as far as I can see offer exactly the same abstractions.
For example Phusion Baseimage is a Docker baseimage that's similar to an OpenVZ container in the sense that it emulates a full running Ubuntu environment. It has is uses but the Docker community rather sees containers that encapsulate a single application with no external processes.
Docker does that, too. Actually, I'm running docker containers as a fast and easy replacement for VirtualBox VMs.
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.