I believe that Tarkin's future significance - and his control over Vader - lies in the role he will play in moulding Vader from the reckless youth of Episode II into the mature, authoritative leader he will become.
Consider how rash and impetuous Anakin is in Episode II. This rashness will ultimately lead to his estrangement from his Jedi allies, his crippling, and the loss of his great love, Padme. After all of these mishaps, he will learn to control his temper and become the ice-cold, calculating Darth Vader.
I believe that Vader accepts his status as an underling to Tarkin in Episode IV because he most likely saw him as somewhat of a role model. Think about it - how else to explain his transition from the emotional, reckless youth of Episode II to the totally calm and in-control Vader? His personality turns around a full 180 degress between Episodes II and IV. Surely the transition phase would have involved learning from someone he respected for their pragmatic, ruthless and expedient style. No one better fits this description than Tarkin. Vader, like all great totalitarian dictators with nothing to fear, would only bow to someone they respect personally, as they can never be constrained by any formal authority. Even if Tarkin does not end up playing a big role in Episode III, I am certain that in the period between Episodes III and IV, Vader will learn to respect and admire Tarkin for being such an effective, ruthless, emotionless operator and this will help mould his character. Having lost so much from being rash in his youth, Vader will learn the value of Tarkin's cold, calculating style. Even by the time of Episode IV, Vader is still learning from Tarkin and being impressed by his ruthlessness and cunning. Consider how impressed Vader is when Tarkin devises the plan to persuade Leia to confess by threatening to destroy Alderaan - a cunning plan that Vader himself was not able to devise.
No wonder Vader looks on Tarkin as a friend. By the time of Episode IV, Vader will rightly be able to say that Tarkin "helped make me the man I am today".