For example, in AD&D 2E a longsword deals 1d8 points of damage to medium and littler creatures and 1d12 points of damage to large or bigger creature. Does 5E have any rules for weapons dealing different amounts of damage based on the size of the monster that's being attacked?
There's not really much else to say. To some extent the simulationism of larger/smaller creatures being easier or harder to hit is baked into their dexterity, AC, or features. (Mostly dexterity, from a casual skim of the MM.)