I'd argue that an MVP is a minimal game with limited features - say, a platform game where you can walk and jump and finish the game, whereas a vertical slice is the complete experience, walk, jump, collect items, fight, achievements, etc, but the story is just a concept and there's only one level.
In other software, MVP is what you can go live with to all of your customers (often replacing something existing and omitting half the existing features to much chagrin).
