The way it works in D&D 3e is reasonably good:
Sling: Simple weapon, 1d4+Str damage, decent range, x2 crit. Slow reload - move action. Also, the sling has easy-to-find ammo - in general you should be able to scavenge up suitable stones.
Light Crossbow: Simple weapon, 1d8 (no Str) damage, better range, 19-20 crit. Slow reload - move action. And the crossbow needs specialised ammo - you can only replenish while in civilised parts*.
Heavy Crossbow: As the light crossbow, but 1d10 damage and an even slower reload.
Shortbow: Martial weapon, 1d6 damage (no Str by default), better range, x3 crit. Fast reload. And the bow requires specialised ammo - you can only replenish while in civilised parts*.
Longbow: As shortbow, but 1d8 damage and slightly better range.
* Okay, a skilled character may be able to build his own. But that takes significant time, which the sling doesn't require.
If you don't have any sort of weapon proficiency system, I would advocate dropping the damage dice of the bows by a step - shortbows to 1d6 (no Str) and longbows to 1d8. If you don't have any sort of improved criticals (whether 19-20 ranges, x3 multipliers, or extra dice on a crit), then I would suggest increasing sling damage to 1d6+Str. And if you allow all missile weapons to add Dex to damage (as I do in my own game... incidentally, the same will apply if you allow them all to add Str, or similar), then I would suggest dropping the slow reload for the sling.
(And let's face it - tracking ammo is a pain. Better to drop that if we can...)
Putting all those options together, I would suggest:
Sling: 1d6+Dex damage, decent range.
Light crossbow: 1d8+Dex damage, better range, slow reload.
Heavy crossbow: As light crossbow, but 1d10 damage and slower reload.
Shortbow: 1d4+Dex damage, better range.
Longbow: As shortbow, but 1d6 damage.
That would seem to give reasonable differentiation - the crossbows are best for pure damage, but they're slow, the bow trades raw damage for range, while the sling is the good middle-of-the-road candidate.
That help any?