#iwanttobelieve that why this guy says is right; so here's some 5e extensions for you.
(Note: this is inspired by Burning Wheel.)
Edit: think about what this means for phalanxes and rogues who get off a successful sneak attack. They stay on the inside …
This only applies if one weapon is categorically longer than the other, like hands < daggers < swords/axes < spears/polearms.
Combat starts with combatants outside the reach of each other's weapons.
While so situated, the categorically shorter weapon has disadvantage on attack rolls.
When you hit on your attack roll, you can choose to stay or move inside or outside the reach of your opponent's weapon.
Moving inside provokes an attack of opportunity made with advantage vs you. On a hit, you stay outside.
If you are inside the reach of your opponent's weapon, the situation is reversed: the shorter weapon has advantage, the longer disadvantage. Note that grappling under these circumstances has no disadvantage and thus grappling or switching weapons may be optimal.
Large monsters should pretty much always count as having the longer weapon in melee. So use missiles or get in close to stab at the soft underbelly, where it'll try to grab you.
Example: Rogue with short sword fighting goblin with spear. They join melee from across the room and this begin outside each other's reach. Goblin attacks, hits, chooses to stay outside. Rogue feels lucky, attacks with disadvantage for the shorter weapon on the outside, hits, and chooses to move inside the reach of the goblin's spear. Goblin misses on opportunity attack, and the Rogue moves inside, and now the goblin has disadvantage to hit; so he drops the spear and grapples.
Why not expand the notion of advantage to damage rolls as well?
Damage advantage is rolling damage twice and taking higher (or lower on disadvantage).
Slashing damage has damage disad vs metal armor but adv vs leather and less.
Wooden / fleshy weapons have damage disad vs metal armor.
I leave the rest as an exercise for the reader.