The Sith care about control of the Galaxy, learning every thing they can about the dark side, and they want to extinguish the Jedi. The inquisitor doesn't care bout all that like Vader and the Emperor do, he just does what he's told. You don't see him trying to find an apprentice for himself to take down Vader, he's cool taking order's. Sith lords are always trying to kill one another.
Although it may be considered a 'Legends' thing now, a Sith lord also had to kill/sacrifice someone/something close/important to them in order to be a sith. Dooku killed Sifo Dyas who he was close friend's with, Vader killed Mace and severed his ties with the Jedi who he looked up to all his life,etc.
And a Sith also has to be under the direct tutelage of the Sith Master,or obviously be the master himself. That's most obvious w/ Kylo Ren not being named Darth_____ right now.