It's pretty clear that Vader as agent of the Emperor was in Governor Tarkin's regional jurisdiction. It pretty much goes without saying then that Tarkin, who was obviously there from the beginning of the Empire, was probably in the important inner circle of the Empire. I doubt either he or Vader considered each other as competitors, but did respect each other's jurisdiction. And in ANH Vader was in Tarkin's jurisdiction.
Dont think Vader fears anyone but the Emperor, but if Tarkin is gonna be part of EP III he would have been around for a long time. So maybe Vader just respect his authority, and maybe they are friends if Vader can have friends. Tarkin might be a vital part of Anakin/Vaders recovery form the wounds he is suppose to get so..............Vader mand Tarkin might have established some sort of friendship with mutal respect for each other