Its been a while since I watched A New Hope but my take on the Vader Tarkin relationship was different. Tarkin was running the operation and Vader was there as the Emperor's eyes and ears and as an advisor. The end of Revenge of the Sith makes it clear that Tarkin & Vader have worked together before and the impression I had was that Vader respected Tarkin to a degree.

When Vader goes to choke the officer Tarkin orders him released and Vader says something along the lines of "as you wish,". Thats a lot different from "yes sir". My take was Vader had made his point to the officers there i.e. don't mess with the man in black and when Tarkin ordered him released he was thinking something along the lines of "ok this is your show we will do things the way you want."

To put it another way its a bit like an Admiral comming on to a carrier/warship as a guest. He may outrank the ship's captain but the Captain is the commanding officer and unless the admiral is supervising him, the admiral may make suggestions and get respect but he is not in charge.

Granted the Captain would not give orders to the Admiral the way Tarkin did to Vader but its the closest example I can think of.

If Tarkin had deciced to take the Death Star to say the capital and try to blow up the Emperor the officers may have gone along with it but that's when Vader's role would have kicked in and no more Tarkin.