AFAIK archery turned out to be the deadliest of the three antique missile weapons. Slings actually did tend to have more power at shorter ranges, and had the nasty habit of battering shields into matchsticks, but in long-range killing power a major bruise and a few broken ribs just don't compare to a yard of wood and a sharp metal point in the lungs.

Sling bullets to the head were nasty, though. Medieval commanders found peasant slingers useful in sieges, and when the Spaniards rampaged in South America the sling was just about the only native weapon they were truly worried about - a hit could kill or blind even through a steel helmet.

That aside, I understand that both slingery and archery are rather demanding pursuits and it takes lots and lots of practice to become proficient in either. Well, peasants chase off beasts and hunt small birds with them, so they get their practice right there, and hunters (usually also peasants) get their archery practice "on the job" too; commanders found both useful.

However, especially with the more powerful bows - longbows and composites - the effective range appears to have been rather greater than even the best slingers could aquire. I have, for example, read that around the Persian Wars the maximum effective range of Persian archers (with composite bows) was about a hundred meters. Well, that worried the Greeks enough that they instituted the sport of 100-meter Dash In Armor specifically to counter it; it paid off in at least Marathon, and probably elsewhere too. Sure, the hoplites were pretty winded after that spurt, but by that point they were in close combat where the archers were understandably reluctant to shoot into (or they might well be in close combat with the archers, an even better result).

Javelins always suffered from poor range and the fact that they flew so slowly open-order troops could easily dodge them; they worked better against close-order troops who had to deal with the javelins poking through their shields as well as they could (armor helped) and soldier on.

Notice that the Roman pila were (well, one of the pair anyway, the other was for longer ranges) specifically weighed for extra impact at the expense of range with a lead ball under the "socket". This plus their very long, thin heads meant they punched through most anything, hence the AP ability in the game. (The Cilician Mercenaries with their harpoons have the same, incidentally; dunno if their range is longer.)

On the other hand, javelins were cheap and cheerful and simple to use, hence their popularity among the poorly-equipped sections of Roman and Greek armies.