If you want distinction, then it should be logical.
By style, not by simple/martial. Mace and hammer are 99% same weapon.
Thy might be, but when they function differently on a mechanical level in d&d they are not
Back n 3.5 a light mace was 1h 1d6x2, heavy mace was 1d8x2 2 handed & the martial light hammer was 1d4x2 but could be used both melee & ranged with a 20ft range increment while a warhammer was a 1h melee weapon that was 1d8x3. in 5e you've only got 1 mace(d6) & everything has the same crit mod but things otherwise convert pretty well for comparison.
If proficiency dice rather than flat proficiency bonus gets used, setting a floor on the roll akin to brutal would be another way of differentiating some weapons from otherHere's some properties that I think could be fun:
Brutal 1/2 : When you roll that number, or below, on your damage dice, you automatically reroll your damage until you roll above said number
Offhand Defensive: When wielding this weapon in your off hand, you gain a +1 bonus to AC (Swordbreaker anyone?)
Twohanded Defensive: When wielding this weapon in two hands, you gain a +1 bonus to AC (iron staff or something?)
High Crit: When you land a critical hit with this weapon, add an extra dice to your damage roll. (might specify the extra die so you could have weapons that do d4 damage, but add d6 when they crit or something)
Swift: You gain advantage to your first attack with this weapon if you drew it as part of the attack action. (Good for katana and hidden daggers)
Two-Ended: This weapon can switch between two damage types whenever you attack with it.
Overwhelming: Whenever you score a critical hit with this weapon, you either push the target 5 feet or the target falls prone.