Attacks are not Spells. +Aether damage to Spells does nothing for things which are not tagged as spells, which require a staff or a catalyst to use by default.

As for why +3-6 Toxic damage out-performs +20% Material, that's because +x% damage increases are all added and lumped together in this game (including the one in the hundreds of % that you get from your 4 main stats) while flat +x damage increases are all added together before being multiplied by everything else. So, going by those screenshots, you seem to have a total of ~500% damage increase to Physical and Toxic (for ~600% total), to which another 20 is only a 3.3% increase; that's about how much your Physical and Toxic are increased by with that amulet. On the other hand, the +3-6 Toxic is added in before that huge multiplier, which is why when you equip that one your Toxic jumps by ~18-36 points, which is a ~6.4% increase to your previous stats.