In addition to what has been said, I think Palpatine knew Tarkin longer than he did know Anakin. And Tarkin has been Palpatine's supporter for a longer time. They were basically a triumvirate after Anakin became Vader and Tarkin always had more political power than Vader. Tarkin's dead created a considerable power shift towards Vader.
(If I had written the prequels, Tarkin would play a much larger role in them.