The difference is this: Short swords are piercing type damage, Scimitars are slashing type damage. So for damage resistance tables they do make a difference, along with what stat gives bonuses for attacking with that style (piercing uses Dex, slashing uses strength). Damage type comes into play as some armors and monsters are resistant to slashing or piercing or blunt, etc.. So for game mechanics this can provide very different results depending on what you are attacking.
Piercing does not always use Dex and slashing does not always use strength. The "finesse" property of weapons is what allows you to use Dex, and both shortswords and scimitars have this property.
They are different damage types, but I'm not sure that there's actually any common monsters that are resistant or vulnerable to one of these damage types and not the other? The only thing I can think of is skeletons which are vulnerable to bludgeoning...
Edit: and I guess the minotaur. But that is because of a magic item...although who knows, maybe we'll see more of that magic item type on future monsters
