I've seen in some videos that the GOG version of the game has some enhancements, such as 16: 9 support and 4K resolution. Is there a place where you can see a list of the improvements that this version includes compared to the original?

I haven't found a full list yet but here's what the community has found so far:

1) widescreen support
2) partial controller support
3) post processing effects that were missing in the original PC release have been added back in
4) cutscenes that were 15fps in the original release are now 30fps

Can anyone confirm or deny these being fixed now, or not?

https://www.reddit.com/r/pcgaming/comments/8z5x7r/silent_hill_4_on_pc_is_missing_at_least_six/

- Post-processing effects aren't missing in the original release.
- Cut-scene are 30FPS like PS2/Xbox now.
- Real Time Shadow is disabled by default now.
- Real Time Shadow still "broken" when you compare PS2/Xbox
- Partial controller support are the same at the 2004 release. (Still difficult to assign all buttons)
- Now game propose 1080p, 1440p, 2160p resolution with widescreen without deformation.
- Still "Dolby Digital" Sounds in options
- "Dolby Digital" sound is just same spatialisation sound on the "Stereo" PS2. Without that, sound stereo is weird. (Like, one channel for a sound come to left for example).
- Still low-rez framebuffer when games freeze for object menu or reading text.
- Low-rez post process
- Noise effect too "small" when you up your resolution.

I keep playing and see if this port have all hauting

"small" is an aspect of dpi/ppi, not of resolution.

Anyway, I can see it on 1440P 27" (=109 dpi).

Check this out
https://steamcommunity.com/app/19000/discussions/0/2941371547493964120/

"Video options include resolution, gamma and brightness... That's it. No texture upscale."

What's this "No texture upscale" BS? Of course the game upscales the textures, or you couldn't play at a higher resolution.
And no, that's not "it", there are more options, like anti aliasing and anisotropic filtering (and more).
Most of this post is just trolling and/or ignorance.

I have read elsewhere that at the very least the closet haunt can still happen in the PC version so take that with a grain of salt.