From his debut in Dragon Ball Z all the way to present times in Dragon Ball Super, Vegeta has been defined by two things, his Saiyan pride and his power level relative to Goku's. For the longest time, Vegeta harbored quite the grudge against Goku even though the mighty Kakarot had never actually beaten Vegeta in a fight. The relationship between two of the last three living, full-blooded Saiyans has grown from antagonistic to just a rivalry, and even now it has become closer to a friendly rivalry. All these years later, the rivalry is still going strong no matter how powerful they become though, which begs the question: What could ever make Vegeta put an end to it?
It's clear that while Goku sees Vegeta as one of his closest friends, the Saiyan Prince still sees him as an obstacle more than anything else. The two can work together and often have each other's back on the battlefield, but at the heart of their relationship is a desire to get stronger. Goku's part in the equation is that he simply wants to get stronger for the sake of getting stronger, while Vegeta specifically wants to be stronger than Goku. If this rivalry is ever going to be put to bed, it will have to be done by the only person amongst the two who is actively making it a rivalry in the first place.
When Goku and Vegeta first fought, the Prince fancied himself a Saiyan Elite while Goku was a low-class warrior, so his inability to quickly mop the floor Goku is what drives him mad to this day. Vegeta was defeated and forced to flee, but not because he'd been beaten by Goku. he was only defeated by the combined effort of Goku, Great Ape Gohan, Krillin, and Yajirobe of all people. In terms of a one-on-one fight with a definitive winner, the record between the two Saiyans is actually 1-0 in Vegeta's favor. Admittedly a dirty move, Vegeta did come out on top of their clash in the Buu Saga of DBZ by tricking Goku into lowering his guard then clobbering him into unconsciousness.
Near the end of the Buu Saga, Vegeta did begrudgingly admit that Goku always has been, and always will be, better than him. However, this humble attitude didn’t last long as Vegeta has gone back to trying to surpass Goku in Dragon Ball Super -- and he actually has a time or two if only briefly. As Vegeta and Goku both step further into the domain of the gods by training under Beerus and Whis in the ways of Destruction and Ultra Instinct, there seems to be no limit to their potential for power. This means that if there's no end to how strong they can become, there will likely be no end to their rivalry, either.
Yet, what would it take for Vegeta to call it a wrap? Defeating Goku wouldn't do it because he's already done that and still keeps trying to one-up his fellow Saiyan. Vegeta has begun his own path to power by studying Destruction under Beerus and leaving Ultra Instinct to Goku and Whis, which was a great move to excise the part of his identity that's always been tied to Goku. But the rivalry is stronger than it's ever been now that they have new, opposing techniques. Goku admitting that Vegeta is stronger likely wouldn't end it either, because Goku did just that at the end of the "Moro" arc in Dragon Ball Super's manga.
Fighting is ingrained into every cell of Saiyan biology, and it's Vegeta's and Goku's Saiyan heritage that will likely be the end of this historic rivalry. Saiyans evolved as a warrior race, so they retain their youthfulness until roughly age 80, but they do get old eventually. Perhaps anticlimactically, the same force that comes for everyone eventually will likely be the only possible end to Vegeta's and Goku's rivalry, Father Time. Once Vegeta and Goku are too old to fight -- which will happen sooner than later if they keep burning away years in the Hyperbolic Time Chamber -- even Vegeta will have no choice but to hang up the bravado (maybe).
