Compared to pure normal weapons, I believe the weapons with inherent bonuses vs. a specific monster type currently do about 5% more damage versus that type and 10% less damage to the other 5 types. And they can't stack as many damage bonuses from other sources before they hit the cap.

Compared to pure normal weapons, I believe special damage weapons are basically the same except with the damage type changed. For example, I believe the Acheron and the Leviathan Blade do the same damage (with the same normal attack pattern) versus shadow-neutral monsters.

I honestly have no clue what they were thinking when they balanced either of these. In certain cases, you might be able to argue that a normal-damage weapon has some inherent advantages that compensate for the lousy damage type and would be overpowered if it became special. But if you're prepared to accept that a straight conversion of normal damage weapons into special damage weapons is balanced, I think converting the "vs. one monster type" branches into special is probably also fine. They already have their base damage reduced enough that I would still recommend most players not use them.