LaserDisc

The 3D Blu-ray has most of the grain kept

The Disney 4K Blu-ray, while improving slightly on resolution, has significant DNR applied; see Blu-ray

LaserDisc is best for stereo.

While Sony 4K Blu-ray has Original Stereo, it is compressed like the 5.1 on the 4K Blu-ray.

The TrueHD 5.1 from the 2008 Sony Blu-ray is free of compression. https://forum.blu-ray.com/showpost.php?p=22484238&postcount=94

But Warner Bros 4K Blu-ray is very similar.

US Image Entertainment LaserDisc is pan-and-scan, but is the best quality release of the film. Widescreen version never released on home video.

US Image Entertainment LaserDisc has the original Dolby Stereo mix.

Germany DVD is a dupe-y transfer ripped from a VHS with German titles, PAL speedup, and mono audio.

UK Arrow 4K Blu-ray > US Kino Lorber/Germany Capelight - caps and audio comparison on this page. Discussion is still ongoing about the 2.0 but as none of the three 4Ks seem to have the true original mix, I think it's fair to leave them all in the blue tier for now and only note that Kino Lorber has the worst PQ.

The initial release of the Capelight 4K mediabook had some playback issues with Dolby Vision on Panasonic devices due to an encoding error, the HDR10 apparently works fine, see Capelight’s official response here. 

The 4K master has a shot at the very end of the movie (on James Caan's face) that looks incredibly soft and not good at all. This was visible on the DCP as well so it might just be a source issue rather than being the fault of Kino, Capelight or Arrow.

The compression is poor, but it still beats the previous Blu-ray. (Luminance plot)

?

The 4K Blu-ray only has Atmos which is possibly the original mix, but the previous Blu-ray’s 4.0 sounds good too.

More Chroma artifacts

The Paramount 4K "restoration" is a desecration. It has completely revisionist color timing, harsh HDR, treatment of stock footage, bad encoding, selective DNR and grain management so bad that the entire screen frequently freezes up with only characters moving around in grain soup. It is so bad that the film's restorer Robert Harris publicly washed his hands of it saying essentially the 2007 restoration (with Willis and Coppola supervising) is how the film was intended and made. This is Paramount's modern version done their way. The new 1080p SDR Blu-rays in print are the crap 4K desecration master with the same problems still there just harder to spot and with crap encodes. Part III fares best of the three and these issues are at their most minimal-but they're still there. The new Coda version is given prominence with lesser encodes for the other two versions. The original mix was remixed into 5.1 decades ago and we have yet another version of this instead of the original Dolby Stereo SR as heard on the LaserDisc release.
The 2008 Blu-ray of the 2007 restoration, while an imperfect outdated disc, is LIGHT YEARS better than this 4K desecration. The only truly major issue is that it is very slightly redder than the 2007 finished master as seen on DCPs.
DFIC review of the hideous crap 4K Blu-rays: https://youtu.be/0uw6-Kcy_UA?si=ob1nDg0wTCvemjH0

Despite being restored alongside the Coppola Restorations of The Godfather: Parts I and II, the Part III restoration is not officially labelled a Coppola Restoration.

Arrow 4K Blu-ray > Criterion 4K Blu-ray > Kino Lorber 4K Blu-ray

Three masterings of the original Dolby Stereo SR mix exist: the original Orion video releases, the Criterion (which is from their LD) and the Arrow. All three are excellent with slight mastering differences between them. The original Orion still is probably the one with the most fidelity, the Criterion is the cleanest with a kind of warm feel to it and the Arrow has a bit of both aspects. The Arrow mastering has a tiny almost imperceptible bit of audio bleeding into the surround that isn't on the Criterion. Each has their own pluses and minuses but you can't go wrong with any of them. Whatever you do though: DO NOT WATCH WITH THE 5.1 REMIX. It is another MGM botch job that fundamentally alters the impact of many scenes. It is vastly inferior to the original mix in every way.

Criterion Blu-ray has exclusive extras, Arrow UHD has the most extras of any release

Dolby Stereo: Arrow 4K Blu-ray or New Line LaserDisc for line absent on VHS, DVD, and Blu-ray releases

Info: Arrow 4K Blu-ray is scanned from interpositive film, thus having a softer than the first title.

• 2015 Lionsgate Blu-ray (but it's missing the extra scenes in the Ultimate Edition)

• 2017 StudioCanal 4K Blu-ray and subsequent 4K Blu-rays are notorius for being heavily denoised and smoothened. Alternatively, 35mm scans for the theatrical cut is also available.

• For the Ultimate Edition with the T-1000 searching John's room and the happy future ending, the 2009 Skynet Blu-ray, which is the older pre-2017 transfer, but has DNR.

Differences of mixes on various releases: see thread and more info here.

2017 Blu-ray (which has the DNRed transfer), the 2009 Skynet Edition Blu-ray, and the 2000 Ultimate Edition DVD have exclusive extras. The 2003 Extreme Edition DVD has some exclusive easter eggs.