In the earliest versions of D&D individual weapon types were barely distinguished from one another, mechanically speaking. All weapons did d6 damage, from daggers and clubs to two-handed swords and lances, at least until Mike Mornard pestered Gary about it. Weapon speed and armor type adjustments came later as well. My beloved Basic/Expert version included variable weapon damage (i.e. daggers d4, swords d8, polearms d10, etc.) as an optional rule, but it is an option that I've always used, as has everyone I've every player with. (S. John Ross doesn't use it, as I recall. But he is exceptional in many ways.)
Anyway, I want to consider here what it would look like if I went back to all weapons do d6. Why would you choose the weapon you choose, assuming no specific mechanical advantages or disadvantages attach to them?
Below are two possible answers, which I think are completely compatible with each other.
[Note that I am ignoring the rule that two-handed weapons strike last in initiative, as that rule is directly attached to the variable weapon damage option (see page B27).]
Answer 1: Aesthetics
From a certain point of view, this is the first, best reason to do anything in an RPG. You're exploring an imaginative space, you might as well populate it with things that come together in an aesthetically interesting way. Taking away specific weapons rules means I no longer feel guilty about choosing a suboptimal weapon (I sometimes worry about not pulling my weight when I'm a player). If I want to, say, bonk the bad guys with a rude wooden club, the d6 rule doesn't actively discourage that behavior.
Answer 2: Narrative-style Advantages
If you are making effective use of your head-brain, there are lots of ways that a specific weapon impinges the gameworld that don't necessarily need mechanics attached. Need a small back-up weapon that can be stored securely but also drawn in a flash? A dagger or shortsword makes a lot more sense than, say, a warhammer. Want to hold off the touch attacks of some creepy ass monster? A spear or a polearm will get the job done a lot better than a club. Need to hack down a door that no one can open? I hope someone is carrying a battle-axe. If all weapons do d6, I suspect it would encourage more people to start thinking about their weapon choices in this way.
Sometimes when you float All Weapons Do d6 rule on the internet, some joker will come back with something like "Well then, I should just spend 1gp to buy 12 iron spikes and hand them out to all my friends. No need to spend the money on anything better, since everything does d6." My response to that is twofold: A) All Weapons Do D6 is not the same concept as All Objects Do D6. Prepare to encounter mechanical penalties for wielding a non-weapon in combat. and B) There's a fine line between clever and stupid and you are are nowhere near that line.