There is a very important difference between Ra's al Ghul and Damien Darhk: Darhk is a sorcerer. As such, he is presumably far more powerful, individually, than Ra's.
The title of Ra's al Ghul is bestowed on a member of the League of Assassins who has shown the necessary skills to lead them. Access to the healing powers of the Lazarus Pit are just a side-benefit of inheriting that title. Being Ra's al Ghul does not, on it's own, grant you any particularly special powers. Note, for example, that Oliver Queen was very close to being Ra's, and Malcolm Merlyn currently is Ra's; neither of them has any kind of supernatural powers.
Darhk, on the other hand, has clearly taught himself magic. The concept of magic didn't even really exist until the current season of Arrow, but between Darhk and Constantine, it's clearly a real thing; this is even beyond the simple mysticism of the League of Assassins: Darhk can "cast spells" that allow him to do things like drain life force and telekinetically control objects.
His magical abilities appear to be supplementing his use of the Lazarus Pit to keep himself young-looking. He practices a form of blood magic, which can heal his wounds without the need for the waters of the Pit. Also, though we've not seen proof of this, one of the more common tropes around blood magic is the stealing of life force from others. As such, it's very likely that Darhk has needed to rely much less on the Pit water's healing powers, and thus, it will remain potent for him for far longer than it did for Ra's.