Area 1: None. You still need the first weapon upgrade in order to destroy the wall monster leading to Area 2.

Area 2: Gunvolt's weapons can destroy the brown blocks, so second weapon upgrade can be skipped.

Area 3: Same as normal.

Area 4: Same as normal

Area 5: Gunvolt can destroy the grey blocks underwater, so the Dive Gear can be skipped.

Area 6: After the control switch and a few rooms full of ice blocks, you can go through a path that is intended to be a shortcut after obtaining Wall Climb. The main difference is that we no longer have to go through a series of tough platforming to skip the cannon shot.

Area 7: No major stuff but you can destroy the brown blocks when starting the fight against Skeleton Boss, so platforms are indeed skippable.

Area 8: If you've skipped the Dive Gear, you will have to exit the tank in order to get through the underwater segments.