Inconsequential, as in, say less than 10% of your normal damage roll. If the base weapon dice + enhancement bonus + whatever is far heavier weighted than the +str mod damage, then I don't see much reason to have the rule - purely for the sake of the damage.
To revisit noob trap, the analog I'm thinking of is Toughness. In 3.5 lvl1, HP: d8 + conmod + 3hpToughness, the feat is significant. Similarly weapon damage 1d8 + 3strmod and getting 3 damage on a miss is a significant percentage. However, once you move up in levels, and your weapon damage is augmented by GMW, Strikes, Smite, Wallop, Power Attack, etc., the miss damage means very little, unless it does trigger auxiliary effects.