Pretty sure the "Editor's Edition" is just a re-release. The Editor's Choice actually appears to include the same extras (mostly level editor tutorials) that the special edition DVD release came with.
I'm sure they are compatible, just install the latest patches.
The editor's choice edition has all of the downloadable maps. It also has a pretty recent patch as well. So if you had a regular edition and an editor's choice edition, you would want to head over to fileplanet or some other download site and pick up the most recent patch and the editor's choice map pack as well as the previous map packs. After you installed those, you would have the same game on both computers.