Hi, everybody!
Coincidentally, I was thinking about possible distinction between these two unit types recently.
And I came to next variant:
1. Rules for light bow infantry should be leaved unchanged: range 3 hexes, roll two dices when stay, one dice when moved.
2. Rules for light sling infantry may be changed this way: range 2 hexes, roll two dices no matter what unit did: stood or moved.
In this variant slingers have shorter range but don't have penalty for movement.
Just remember how slingers were depicted in Alexandr movie, fast and shooting on move.
Maybe it make the sense, what do you think?