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
They both are very different. Docker is lightweight and uses LXC/libcontainer (which relies on kernel namespacing and cgroups) and does not have machine/hardware emulation such as hypervisor, KVM. Xen which are heavy.
Docker and LXC is meant more for sandboxing, containerization, and resource isolation. It uses the host OS's (currently only Linux kernel) clone API which provides namespacing for IPC, NS (mount), network, PID, UTS, etc.
What about memory, I/O, CPU, etc.? That is controlled using cgroups where you can create groups with certain resource (CPU, memory, etc.) specification/restriction and put your processes in there. On top of LXC, Docker provides a storage backend (http://www.projectatomic.io/docs/filesystems/) e.g., union mount filesystem where you can add layers and share layers between different mount namespaces.
This is a powerful feature where the base images are typically readonly and only when the container modifies something in the layer will it write something to read-write partition (a.k.a. copy on write). It also provides many other wrappers such as registry and versioning of images.
With normal LXC you need to come with some rootfs or share the rootfs and when shared, and the changes are reflected on other containers. Due to lot of these added features, Docker is more popular than LXC. LXC is popular in embedded environments for implementing security around processes exposed to external entities such as network and UI. Docker is popular in cloud multi-tenancy environment where consistent production environment is expected.
A normal VM (for example, VirtualBox and VMware) uses a hypervisor, and related technologies either have dedicated firmware that becomes the first layer for the first OS (host OS, or guest OS 0) or a software that runs on the host OS to provide hardware emulation such as CPU, USB/accessories, memory, network, etc., to the guest OSes. VMs are still (as of 2015) popular in high security multi-tenant environment.
Docker/LXC can almost be run on any cheap hardware (less than 1 GB of memory is also OK as long as you have newer kernel) vs. normal VMs need at least 2 GB of memory, etc., to do anything meaningful with it. But Docker support on the host OS is not available in OS such as Windows (as of Nov 2014) where as may types of VMs can be run on windows, Linux, and Macs.
Here is a pic from docker/rightscale :