Edge versus Edge-V

How are the Edge and Edge-V different? If their intended purposes differ, then how? Do they both use the same case? Or do they have different cases? How do they relate to the Captain board?

Hello, In addition to having different form factors, differences exist in I/O and connector layout. The Edge-V is largely self-contained, a single board, though an optional extension board, the M2X, is available to mount an M.2 NVMe SSD, also offers POE.
The Edge is a System on a Module(SoM). An optional base board is available for the Edge, called the Captain. The Edge mounts(docks) to the base board. The Captain adds many functions, M.2 NVMe SSD mount and I/O to the Edge.
A specification comparison chart for the Edge family is here.

The Khadas DIY case can work with the Edge-V(without M2X), albeit with a small modification, see here.
For the Edge, or at least Edge+Captain, the Khadas Kap Case is available, it includes a mount for the TS050 display+touchscreen. I do not have the TS050, I use the Kap Case to house the Edge+Captain, the display mount can be left off of the Kap.
For more detailed information on the Edge, Edge-V and accessories, see here.