The Plagued and Pure Tears are essentially the same item, seeing as you cannot possess both at the same time. The Pure Tear can be obtained by unlocking a new trade with a villager priest at night (this is a rare trade, you can ONLY unlock it at night). The Plagued Tear can only be obtained as a rare drop from Ghasts in the Nether.
Here's the catch: once you leave the Nether, all Plagued Tears you have will be transformed into Pure Tears! This as the same with Pure Tears entering the Nether; they will be changed into Plagued Tears. Because of this transformation, potions amplified to the third level (III) can only be brewed in the Nether, while potions enlongated to the third level (Very Long) can only be brewed in the overworld.
They are used in every potion's brewing process, as every potion can either be enlongated or amplified to a third state.