Arrow hits armoured guy in chest at max bow-range: -arrow bounces off armour- "Pfff!"
Stone hits armoured guy 2 in chest at max bow-range: -sound of wheezing-
Generally speaking, you'd want to get close to an armoured opponent if you are armed with a bow in order to have a good chance of penetrating through armour, especially in regards to more ancient bows, where I believe draw-weights tended to be around ~70-100 pounds, with thinner arrow shafts than in later periods (To the general exclusion of horse-archers, who tend to prefer lighter arrows even in later periods).
However, a stone against an armoured man who might not be wearing a padded garment... is going to have a bad day. With padding garments, it is harder to say.
So... an armoured slinger ought to be feared by bowmen, even if said bowmen were armoured!