Come to think of it, doesn't the Mesopotamian version have energy drain like all succubi? How come Lilith doesn't have that skill?
Arccoding to the wikipedia article on Lamashtu it was less energy drain and more devouring your flesh and more; "Her evil deeds included (but were not limited to): slaying children, unborns, and neonates; causing harm to mothers and expectant mothers; eating men and drinking their blood; disturbing sleep; bringing nightmares; killing foliage; infesting rivers and lakes; and being a bringer of disease, sickness, and death."