Re3 (reverse engineered GTA III) makes the game playable on modern systems. It includes many bug fixes and improvements (eg. widescreen support, free mouse look, neo reflections...).
Topic on Talk:Grand Theft Auto III
Keep in mind that linking to any mirror for re3 or reVC on the wiki should be avoided considering Rockstar/2K took down the project.
Almost everything that is possible in re3 is possible in the base game via modifications like Skygfx, SilentPatch, ThirteenAG's widescreen fix etc.
Visually, you can get similar results with the stated mods/scripts. Only with re3, the game felt way smoother and responsive on my system. Mods like the Tri-Pack and natural HD textures can be used to enhance the visuals.
I strongly believe re3 should be mentioned on the page, but agree it's not a good idea to link to downloads while its legal status is being questioned (if that litigation is even still ongoing).