Fizzlebeef: After thinking about this some more, I'm not sure why I needed to install Wine to play the game. The GOG distribution includes a "wine_standalone" directory. I'm not sure what the point would be of including this directory if it doesn't work or if you have to install Wine yourself anyways.
Anyone have any idea?
As I said above, it's probably a matter of missing 32-bits libraries. GOG bundles wine, but it doesn't bundle all the 32-bits libraries required by wine, since for some of them (like graphic drivers) it depends of your specific setup.
When you installed wine, it probably installed as a dependency the 32-bits libraries required by wine, and GOG's wine did use those too.
Since I don't have any details, this is just a wild guess, but it seems a very likely scenario to me. And it does make sense - GOG bundles what's required to run the game like it were a native game. For native 32-bits games on 64-bits systems, GOG requires you to manually install the 32-bits libraries, the same goes here.
As for the "no such file or directory" message which may trouble you, it's actually what happens when you try to run a 32-bits program on a 64-bits system without the compat libraries : when you run an ELF binary (ELF is the native binary format of Linux, like .exe for Windows) it in facts runs the dynamic linker, /lib/i386-linux-gnu/ld-linux.so.2 for 32-bits or /lib/x86_64-linux-gnu/ld-linux-x86-64.so.2 for 64-bits, and if /lib/i386-linux-gnu/ld-linux.so.2 is missing, it'll just say "no such file or directory".