Weapon length and range is an obvious non-damage-based difference between weapons, but some not-so-obvious distinctions can also come into play on an ad hoc basis. For example, if two characters are throwing daggers at an enemy, but one is throwing a dagger balanced for throwing, that character should get a bonus on the attack. Another example: if a character has a weapon that is designed to be effective at getting through plate armor, attacks with that weapon should be at a bonus when used against plate armor.
The second example was the idea behind the weapon adjustment tables, but I'm talking about an easier, more laid-back approach. A player says, "I want a weapon that's effective against plate armor." The GM says, "OK, here's a specially-designed polearm that costs 7 gp, or you could get a mace for 8 gp." If the group is really into weapon porn, or if the DM uses the weapon adjustment tables as a guideline, things can get more specific (lucerne hammer, footman's mace;) otherwise, you can just make it up. The player writes "maces are effective against plate armor" and gets a bonus when attacking opponents wearing plate armor. I'd only give a +1 for such a weapon, because I work on the principle that if one side has a greater advantage than the other, they get a +1, or a +2 if there are several advantages, a +3 at the most if there is a great advantage. I keep it loose and don't split hairs.
To me, this approach meshes well with the original approach of a single die roll representing several attacks and maneuvers in combat. Hardly any monster gets multiple attacks, one obvious exception being a hydra, which for all intents and purposes is more than one monster, hence the multiple attack rolls. I'd give a chimera or two-headed giant multiple attack rolls as well, but other monsters would just get a bonus to their attack roll if they can attack more times than their opponent. Likewise, a weapon that in theory should allow multiple attacks should get a +1 bonus against an opponent unless that opponent can specifically counteract that kind of attack. A short sword with a double-edged blade that could slash on the backstroke might get a +1 compared to a single-edged blade like a machete, for example.
So in general: any intrinsic quality of a weapon that gives an advantage in a specific combat situation gives a +1 to the attack roll. Different weapons get bonuses in different situations, which is why different weapons exist.
This approach can apply to more than just attack bonuses, as I'll explain in future posts.