I assume it's the same thing though all in all, but just thought I'd mention :D

You had it right with the idea of "they're technically different games", actually. ^.^

The 2006 Special Edition adds additional features to the original 2005 DMC3 release, and isn't a rerelease (in the remaster/remake sense) in and of itself. Thus, as with the other games in the 2018 HD Collection, the version of 3SE included in it is an HD rerelease of a much, much older game.

To list the differences more directly, the HD collection version fixes the porting issues of the original release [see the game's negative reviews], and- as with the other games in the collection- is stated to improve the native resolution from 720p to 1080p (note, this'd be what the game upscales from, meaning the graphics should be noticably more detailed the larger your actual screen resolution is) and adds widescreen support.

In addition, textures are supposed to be sharper and character models are supposed to be better detailed, and there's supposed to be bonus content (eg, concept art) included in the collection. Finally, some technical issues with certain non-gameplay elements (eg, game menus) from the original release have supposedly been fixed.

While the graphic overhaul and overall polish improvements from the HD Collection are a loss when considering the original release), the gameplay-affecting porting issues on this release can apparently be fixed with some unofficial patches and a bit of effort.

Further, on the positive side of things, this version apparently runs far better on low performance PCs, has less input lag in general, and has a cheat code that allows for quick-saving that apparently wasn't carried over to the HD version. So for some, this may nevertheless be the superior version, or at least may be considered as being reasonably comparable.