What is the difference between a game ban and a VAC ban in Steam?

The only difference is that a game ban is given by the game developer and a VAC ban is given by the Valve Anti-Cheat system. Also a VAC ban can be given for more than one game (usually all games using the same engine).

• VAC bans are given by Valve's anti cheat system and either automatic or given by employees.
• Game bans can be given by custom technologies, users or game devs themselves (such as CS:GOs Overwatch, which leads to a game ban after a community vote based on replays)
• Game bans can be expanded to VAC bans if the user in question is being caught by VAC.

Game bans must at all times uphold the VAC rules, for example a game ban must not ban the player from playing the game offline. So basically a game ban is the same, just given by a different system and likely for different (more) reasons and not just for cheating.

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.