CentOS 6’s kernel is too old to usefully run Docker, even the old versions where it’s nominally supported. Some official CentOS kernels broke Docker when we were still actively using it.
CentOS 7 works fine, but if you use docker run -v to share host content with containers, you will forever be fighting with SELinux.
Ubuntu 16.04 works fine.
Beyond the SELinux bit and host-system firewall concerns, I haven’t noticed any significant differences between CentOS and Ubuntu. Taking our Docker-based system that ran on CentOS 7, it ran almost unmodified on Ubuntu. (All of the differences were getting prerequisite packages installed, filesystem partitioning, that sort of thing; the “launch a dozen containers in order” automation worked fine.)