Elemental Weapon Damage ONLY affects the white damage (melee damage) of a weapon while you have a buff that shows up on your Stance Wheel . Confusingly, this has NO AFFECT on the elemental damage (colorful damage) of your buffed melee attacks.
Elemental Damage ONLY affects colorful damage numbers, but DOES NOT affect tick damage. Purity, Corruption, Paralysis, and Poison ARE NOT AFFECTED by this stat.
I spent about two hours in the dojo today whacking dwellers trying to figure out exactly how Elemental Damage and Elemental Weapon Damage special effect stats are different and what exactly they will affect.
Elemental Weapon Damage will affect the MELEE damage of a weapon that is under the affect of an elemental buff such as Fire Talisman, Purification Talisman, or Galnut Broth but WILL NOT affect the elemental damage of your melee attacks while buffed. Weapons with an Imbue [Element] stat won't be affected by Elemental Weapon Damage. As a general rule of thumb if you have a buff that shows up on your Stance Wheel your white damage from melee attacks will be increased by your Elemental Weapon Damage stat for the duration of the buff. Elemental Weapon Damage does not appear to affect white damage from any other source such as soul cores.
Elemental Damage has no affect on melee attack damage. This stat affects elemental damage jutsu like Fire Shot Talisman, Water Shot Talisman, Yarokoa Feathers, etc., explosive affects such as the fire damage from bombs and Fire Shuriken, and elemental damage dealt by Yokai Abilities such as Yasha, Nue, etc. As a general rule of thumb if you are dealing fire, water, or lightning damage regardless of the source of the damage the colorful damage numbers will be increased by your Elemental Damage stat EXCEPT for the tick damage dealt by the Scorched (fire) status affect. Purity, Corruption, Paralysis, and Poison are not affected by this stat.
Only the weapon with the stat on it is affected, so you need to have the weapon equipped and be using it to get the benefits of both of these bonuses (you won't get the benefits for equipping it in your offhand).
Did some more testing and found that you will in fact get both stats from both weapons that are equipped regardless of whether you are using it at the time.
Hopefully this helps some people square away their gear for different builds/needs.