There really is no difference. For almost all intents and purposes, GOG’s offline installers can be treated the same way as physical CDs of way back then, with one of the only exceptions being that you cannot resell them.
Nowadays the Heroic Games Launcher is the preferred solution for downloading and running GOG games. It’s a community-run project, but officially affiliated with GOG.
Have you tried the Heroic Games Launcher? It’s currently the most advanced launcher for GOG games on Linux, even implementing support for some of GOG Galaxy’s online services. Additionally it’s officially affiliated with GOG, so you can “donate” to the project by purchasing games through the client.
How did you try to get The Thaumaturge to run? I found that I could run many games on my library without issue using the Heroic Games Launcher, which is arguably the premier Linux client for GOG games.