"Two-handed mighty weapons come in many forms, from clubs the size of tree trunks to greatswords and axes seemingly made to be wielded only by statues. When these enormous weapons are swung by human hands, stories are made."— Game Guide(src)
Two-Handed Mighty Weapons are a heavier variant of Mighty Weapons, only usable by Barbarians. These are oversized, two-handed Axes and Maces (rarely Swords), seemingly too large for an average human. All of them, regardless of how they look, have attack speed of axes.
Two-Handed Mighty Weapons can be freely transmogrified into two-handed axes and maces, but only Barbarians can do the other way.