The great virtue of slings is that a lot of pastoral cultures had lots of people with the skill to use them. They require no material except leather, and can use stones for ammunition (though cast-lead bullets were popular for ammunition), so any society that tended and killed animals could produce them cheaply. This more than offset the fact that they are a fairly difficult weapon to use - shepherds and the like often had plenty of time to practice, and regular need of a cheap and reliable ranged weapon to drive off animals that might prey upon their flock.

In terms of damage, the term "bullet" used for the purpose-made ammo is instructive. While significantly slower than a bullet from a gun, sling bullets were quite heavy, giving them quite impressive energies. I've seen some modern tests that suggested energies similar to modern pistol rounds, which is significant. Being rather larger and slower would affect the wounds - they aren't going to make the same kind of neat little hole, obviously - but that's still a very heavy hit. Many recovered sling bullets are heavily deformed from impact, and there are written sources suggesting that this was so common that the ancients believed that the rounds were actually melting in flight (this is impossible - modern experiments with firearms show melting only at extreme velocities far beyond the reach of any muscle-powered weapon).

In accuracy and range, there's sources to suggest that slings were comparable to bows as well. The real killer is that if you weren't recruiting men who'd used slings from boyhood, it took far longer to train a slinger than it did an archer.