Slingers were actually the ones with greater range and penetrative power, if they could use lead shot. Our main evidence for their effectiveness is Xenophon's account of the March of the Ten Thousand:

"Now I am told that there are Rhodians in our army, that most of them understand the use of the sling, and that their missile carries no less than twice as far as those from the Persian slings. For the latter have only a short range because the stones that are used in them are as large as the hand can hold; the Rhodians, however, are versed also in the art of slinging leaden bullets." (...) And the barbarians were no longer able to do any harm by their skirmishing at long range; for the Rhodian slingers carried farther with their missiles than the Persians, farther even than the Persian bowmen.

-- Xenophon, Anabasis 3.3.16-17, 3.4.16