The differential pressure controller is found in Vault 11, and is used for repairing the air filtration system in Brotherhood of Steel's hidden bunker in Hidden Valley for the Still in the Dark quest.
The differential pressure controller is located in one of the first rooms one may encounter on the lowest level of the vault, flooded with water (note there are air pockets near the ceiling, though). It will be stashed in a locker and appears before the quest has been initiated. This room is NOT one of the rooms in the reactor area. It's not far from the overseer's office and is in a single locker.