Making your D20 roll more meaningful
There is a very good skirmish game called 'Rangers of Shadowdeep' that uses something similar to this on an opposed roll basis. If my memory is correct both players roll a d20 and the damage is the difference. between the two after modifiers.
It's worth checking out, though IMO systems like these become a bit too complex because of the numbers involved.
The system that I very much enjoy is the 'Warmahcine' system (another wargame), where you have a Defence (How hard to hit), and an Armour (how tough) stat, and when you roll damage all weapons rolls 2d6 and it's the amount over the armour that is inflicted. The numbers are a bit more manageable in this kind of system.
My slight alteration to this rule is to have the number of success levels on an attack roll equate to the minimum number that can be rolled on the damage dice. For example, if you require a 13 to hit, but roll a 17, that's 4 success levels, so when you roll your D8 long sword damage you treat a result of 1-4 as a 4. This is capped at the weapons maximum damage, so 5 success levels with a dagger strike would still only result in 4 damage. I just got fed up with rolling a 19 to hit and thinking wow, what an awesome strike, to then roll a 1 on the damage dice. It didn't seem right after such a "solid" hit.
I do something similar with less math. (*Basic) In my games I oblige the Fighter’s archetype to function mechanically fittingly more lethal & heroic. Expecting combatants respect fighters veracity on the battlefield every attack even a "miss" fighters make their opponents pay for engaging them in combat, inflicting their ability score bonus in damage per the weapon type in melee. Speeds up combats and feels right for the player role-playing the fighter and subclasses. (*Advanced) as the fighter levels say at 3rd I'll also have the critical to-hit-roll tick down for fighters. It's smoother more abstract, than a giant list of attack Proficiencies / Feats.
Allows the player to describe the attack however they want. (*Expert) 5th Cleave the leftover damage from a kill, rolls over to the next closest combatant in melee player’s choice. (*Immortal) 10th add Fighter’s level to each to each attack hit or "miss" in melee.
Depending on the game you play if it has skill advancement add them according to your table's fun.