Every stage can be played on Normal, Hard and Nightmare. Hard opens when you beat the stage on Normal; Nightmare, when you beat it on Hard. Each difficulty win grants its reward once (see Quests).
What changes
Shooters (Archer, Carnivorous Plant, Lantern Ghost) only appear from Hard on, up to 7 alive at once on Hard and 9 on Nightmare.
Bosses: on Normal they only use their first skill; on Hard and Nightmare they use both, and more often (25% shorter cooldown on Hard, 45% on Nightmare).
Nightmare: monsters gain +5% HP for every minute that passes.
Boss drops: the chance is higher on Hard and much higher on Nightmare. The guarantee (hunt) is the same on all.