While Vader might dramatically physically outclass Tarkin, both of them are tactical geniuses in their own right. Moments like this prove why Tarkin deserved his place in the highest echelons of the Empire and show why Vader would take orders from him.
Why did Grand Moff Tarkin have more authority over Darth Vader than other Imperial officers, even though he wasn't higher ranked than Vader? Grand Moff Tarkin did not have authority over Darth Vader; he had authority over the Death Star, and so was formally running the meeting that most likely generated this question.
The Sith, Darth Vader and Darth Sidious, lead the Empire, thus they outrank everyone else. In terms of the Military Rank, Vader, Tarkin and Thrawn (Possibly the other Grand Admirals and Grand Moffs as well) were all Supreme Commanders, the highest rank. Palpatine was the only one who could override their orders.
Vader Outranked only by Sidious - CON: The only thing I can find that says Tarkin outranked Vader is a quote by David Filoni in 2015 during an interview about Rebels. He actually says that Tarkin outranks Vader.
What Darth Vader Did RIGHT AFTER The Empire Strikes Back (CANON)
Who can beat Darth Vader easily?
1 Obi-Wan Kenobi
Obi-Wan faced off against Darth Vader four times and physically beat him twice while beating him through his Force development in a third fight. Obi-Wan Kenobi had the high ground that led to Vader's disfigurement and becoming the ultimate Star Wars cyborg.
A Grand Moff received drastically increased authority, military forces, and funding as fitting their enhanced position. The position of Oversector governor was the sixth highest in the power structure after the Emperor, Supreme Commander, Grand Vizier, Advisors, and Grand Admirals.
In terms of Palps, Tarkin knows Vader is Sith, they talk about using the force in ANH, when Vader chokes Tagge. Also, Vader is always waving his red saber around.
However, once Luke fully embraced his power in Star Wars: Return of the Jedi, he proved that he was indeed stronger than Darth Vader and struck his father down.
Vader knew that he could never defeat Palpatine by himself (especially after his master bested him on Exegol as well), which is why he tried to recruit Luke to help overthrow him during the Empire Strikes Back. He may have not been able to overthrow the Emperor, but he could throw him over the railing to save his son.
Legends-era Grand Admirals outranked Grand Moffs, but were outranked by the Grand Vizier (Sate Pestage), the Supreme Commander and Executor (both ranks held by Darth Vader), and the Emperor himself.
Infil'a put up a fierce fight against Vader, almost killing him by throwing the already damaged Sith Lord off of a cliff. Despite severe damage to his suit, Vader survived the fall, and repaired himself with parts from Infil'a's droid, Arex.
He was the governor of the Outer Rim (Tarkin that is), but could he annul Thrawn's orders or did he outrank Thrawn. Tarkin probably outranked Thrawn since he was a Grand Moff (a political rank) whereas Thrawn was a Grand Admiral (military rank).
Yep. Darth Maul didn't really say or show straight up that he "feared" Darth Vader, but he openly stated his belief that he could not defeat Darth Vader without the help of the Jedi, therefore on his own.
The only Sith Lord who actually killed Vader, Darth Sidious kept his apprentice in line for over two decades. His true advantage was Force lightning, a power which would wreak havoc on the suit keeping him alive.
