A bit of a difference here. From what I recall, a VAC Ban only affects games that actually use Valve’s Anti Cheat, hence the name, which is not so popular anymore as far as I know. And they’re supposed to only affect that game individually (but since they’re public, other game/server owners can decide to ban you too).

Getting an Xbox Live or PSN ban will prevent you from playing any game that relies on the respective service’s network (which they enforce for any game that has online multiplayer in it).