Equality versus priority: A useful distinction

Both egalitarianism and prioritarianism give value to equality. Prioritarianism has an additively separable value function whereas egalitarianism does not. I show that in some cases prioritarianism and egalitarianism necessarily have different implications: I describe two alternatives G and H such that egalitarianism necessarily implies G is better than H whereas prioritarianism necessarily implies G and H are equally good. I also raise a doubt about the intelligibility of prioritarianism.