Lethargy is the name of the state: that heavy, sleepy lack of energy and interest. If lethargic is how you look or feel, lethargy is the feeling itself. You can sink into lethargy after a long illness, in midsummer heat, or when a job has drained you. In everyday talk it means you have no spark. In medical English it is stronger: a serious drowsiness, a warning sign, not just a slow Monday. It is usually uncountable — you have lethargy; you do not often “have a lethargy.” The word is a little formal. It comes from Middle English litargie, from Medieval Latin litargia, from Late Latin lēthārgia, borrowed from Greek lēthārgíā “drowsiness,” from lḗthargos “forgetful, lethargic” plus a noun ending. The river Lethe sits in the background of that root, but lethargy now is about energy, not about wiping memory.