I've always been keen on the idea of weapons having different effects. It's a hell of design challenge though! Giving every weapon a reason to exist without making your game tedious.
Maybe we'd call them 'Expertise effects,' which would trigger if you had advantage and both attack rolls hit. Certain classes could get access at higher levels. But the problem is that 5e doesn't have many things you can do to your opponent other than damage that aren't, well, really strong.
In ZEITGEIST, we have shotguns do extra damage if you have advantage and both dice hit. Maybe picks could do the same in 5E.
But what else do you have? Moving (like if you slam them with a hammer) or tripping (like with a hooked polearm) isn't as meaningful as in 4e.
Grab someone with a whip?
Impose disadvantage on the enemy's next attack with your excellent swordplay? Blind them by slashing above their eye? Do you want to include the ability to actually wound, like gouge eyes, rip throats, and cripple limbs?
Maybe let you cleave into another enemy (with a greatsword or glaive)?
PF2 has some leeway because of the multiple action system. Normally attacks after the first get a -5 penalty, so you can differentiate a rapier - which if you stab the same guy multiple times you get only a -4 penalty - from a scimitar - which is better as slicing
different guys.
Then again, maybe differentiation isn't that important. How often do PCs switch weapons even in the course of an adventure, let alone a fight? Where weapon differentiation would feel meaningful is if you had monsters be designed so you needed different weapons to handle them. AD&D had weapon-vs-armor-type tables, but that provided non-standardized numerical modifiers to AC, which were complicated and hard to remember. But 5E offers a way to make it simpler.
Sturdy Armor: This creature has resistance to weapon damage. When it is hit by a two-handed bludgeoning weapon, the armor cracks and it loses this ability.
Wary Shield: This creature has resistance to weapon damage, but after an attack hits it, this ability doesn't apply to other creatures that attack it before the end of its turn, because it can only guard against one.
Dancing Serpent Stance: This creature has resistance to weapon damage from attacks that aren't Dexterity-based. If it is grabbed or falls prone, it loses this resistance for a round.
Folded Crane Stance: This creature has resistance to damage from piercing weapons.