It's not that it's any different, it's that it's standardized. The idea is that a Docker container would be portable between different PaaS hosts (and from your own staging environment to those hosts!) without rebuilding, because they'd all be using the "Docker standard for deployment."
A PaaS host saying they supported Docker would imply that they'd be using, for example, SquashFS for container format, AuFS instead of OverlayFS for union-mounts, LXC instead of OpenVZ/Xen/KVM for isolation, and any other set of things your container might subtly rely upon.
The culmination of this, I imagine, would be a PaaS host allowing you to specify the "stuff" you want to run just by the URL of the container-image.
A PaaS host saying they supported Docker would imply that they'd be using, for example, SquashFS for container format, AuFS instead of OverlayFS for union-mounts, LXC instead of OpenVZ/Xen/KVM for isolation, and any other set of things your container might subtly rely upon.
The culmination of this, I imagine, would be a PaaS host allowing you to specify the "stuff" you want to run just by the URL of the container-image.