Back in those ancient days of yore when giants walked the earth and D&D was O, every weapon and every monster did 1d6 damage with a successful attack. The rationale was, I believe, that because a dagger was capable of killing a person with a single blow, and a two-handed sword was equally capable of inflicting a mere scratch, that having different damage ranges for different weapons would be functionally pointless, though I believe (I'm not really sure here) that weapons like greatswords got to roll two dice and took the best score. Maybe not. Anyway, the system is still used by people playing with the original D&D Little Brown Books, and with S&W White Box. Somebody on the Swords & Wizardry forums suggested using it for S&W Core Rules as well, but using a d8 instead of a d6.
I find the idea attractive for one major reason: it makes a character's choice of weapon largely aesthetic; there's no weapon-damage min-maxing to be done. If a scimitar suits a character conception better than a greatsword, the player won't be tempted to go for an inappropriate weapon for mechanical reasons. Also, there's nothing for the DM to remember when it comes to what damage a monster does with its fangs or claws or tail-bash or whatever.
It does have its aesthetic down-side though. In a game in which encumbrance is actually important, everyone is going to be using daggers exclusively — and why not? They're lighter, smaller, you can throw them, and they do just as much damage as anything else.
The most important reason not to adopt Ye Olde Damyge Systeme though is this: it reduces still further the use of all those excellent funny-shaped dice. And the poor d12 already gets the short end of the stick there.
I think I'll stick with the many-dice model.