Primarily the PS4/PC version has pre-rendered backgrounds and updated textures for full HD (1080p) while the GameCube version is 480p. The update also runs at 60fps on PC. Other than there there was a new easier difficulty mode in the PS4/PC release, a new control scheme, some bonus costumes, a widescreen and Pan & Scan modes, and achievements.
IGN has a list of differences between the original, Remake on GameCube, and the PS4/PC HD Remaster: Resident Evil Version Differences