First off, VAC ban in case of some games covers a wide group of games - you get banned in one, you're banned in all. Besides that, VAC ban applies only, and strictly to cheating in the game through modifying the game behavior - using external tools or changing the game code. It's applied automatically through an external tool that analyzes game integrity during gameplay, discovering modification to the game code.
Developer ban AKA game ban only applies to a singular game in which you have cheated, and applies to any notoriously disruptive players - griefing, cheating in competitive games through abuse of in-game mechanics, severe violations of chat etiquette etc. It can be applied manually, or through dedicated tools built into the game.