The GOG version has v-sync, gamma correction, antialiasing, integer scaling and other stuff. Also has easier controller setup. Other bugs and stuff have apparently been fixed.
Classic REbirth on the other hand has resolution options and texture filtering. It also has some kind of rumble support, not sure how accurate it is, but it's nice to have. It also supports gameplay changes like quick turn and tactical reload. Classic REbirth is also based on the latest versions of the Japanese PC ports, but I'm not sure if they really are better, since the GOG versions has been patched a lot.
Some people are saying that Classic REbirth has better audio quality.
Hopefully we will get a version of Classic REbirth, that eneables us to have both GOG fixes and REbirth fixes combined.