I keep rereading the Docker documentation to try to understand the difference between Docker and a full VM. How does it manage to provide a full filesystem, isolated network
I like Ken Cochrane's answer.
But I want to add additional point of view, not covered in detail here. In my opinion Docker differs also in whole process. In contrast to VMs, Docker is not (only) about optimal resource sharing of hardware, moreover it provides a "system" for packaging application (preferable, but not a must, as a set of microservices).
To me it fits in the gap between developer-oriented tools like rpm, Debian packages, Maven, npm + Git on one side and ops tools like Puppet, VMware, Xen, you name it...
Why is deploying software to a docker image (if that's the right term) easier than simply deploying to a consistent production environment?
Your question assumes some consistent production environment. But how to keep it consistent? Consider some amount (>10) of servers and applications, stages in the pipeline.
To keep this in sync you'll start to use something like Puppet, Chef or your own provisioning scripts, unpublished rules and/or lot of documentation... In theory servers can run indefinitely, and be kept completely consistent and up to date. Practice fails to manage a server's configuration completely, so there is considerable scope for configuration drift, and unexpected changes to running servers.
So there is a known pattern to avoid this, the so called immutable server. But the immutable server pattern was not loved. Mostly because of the limitations of VMs that were used before Docker. Dealing with several gigabytes big images, moving those big images around, just to change some fields in the application, was very very laborious. Understandable...
With a Docker ecosystem, you will never need to move around gigabytes on "small changes" (thanks aufs and Registry) and you don't need to worry about losing performance by packaging applications into a Docker container at runtime. You don't need to worry about versions of that image.
And finally you will even often be able to reproduce complex production environments even on your Linux laptop (don't call me if doesn't work in your case ;))
And of course you can start Docker containers in VMs (it's a good idea). Reduce your server provisioning on the VM level. All the above could be managed by Docker.
P.S. Meanwhile Docker uses its own implementation "libcontainer" instead of LXC. But LXC is still usable.