Vegeta is probably one of the most popular characters in the Dragon Ball ZBut he's always been limited by his desire to become stronger than Goku. However, there's a clear reason why that never happens, and it has to do with how Vegeta thinks, not just about himself, but about Goku as well.
Vegeta has long been haunted by the fact that Goku is still stronger than him. As the Prince of all Saiyans, Vegeta feels that strength is his birthright, and every time Goku surpasses him, it hurts his pride. After all, in his view (at least initially), Goku is just a lesser Saiyan with no right to the power he holds, and Goku constantly displays insane new levels of power that seem to flaunt this fact in Vegeta's eyes. But this explains why Vegeta will never surpass Goku, and it's perhaps the most heartbreaking aspect of his character.
Vegeta Doesn't Have the Mindset to Become Stronger Than Goku (At Least in Dragon Ball Z)
There is a big difference between Goku and Vegeta when it comes to how they approach getting stronger. Goku was never motivated by the desire to be stronger than others; his goal is, and always has been, to become stronger for himself. Vegeta, on the other hand, is almost solely motivated by the desire to surpass Goku.. He is haunted by this, and every step Goku takes is a blow to his self-esteem.
That's not to say that Goku never compares himself to others, especially the villains he has to defeat, but Goku doesn't judge himself and his progress solely by comparison. For Goku, the goal is always to become stronger than he was yesterday, to the point where fighting powerful enemies is more of a means to an end than an end in itself. Basically, Goku has never cared about becoming stronger than Vegeta the way Vegeta does with Goku. Instead, Goku's goal is self-improvement, in line with his martial arts intellect.
Throughout Dragon Ball Z, Vegeta has rarely been interested in self-improvement for the sake of self-improvement. It has always been to beat Goku, to feed his ego with superiority. It is not enough for Vegeta to be stronger than yesterday, as long as he is still weaker than Goku. This mindset of his holds him back and prevents him from growing in a natural way. If Vegeta could put aside his obsession and grow for the sake of self-improvement, he could discover levels of power that even he never thought possible.
Vegeta's jealousy and ego have always been his biggest obstacle, not Goku's.
The best example of the difference in their mindsets is easily the beginning of the Majin Buu Saga. Vegeta cares so much about surpassing Goku that he is willing to make a metaphorical deal with the devil to gain power, just for the chance to defeat Goku. Even with the life he has built on Earth and the family he has now, Vegeta is willing to give up everything for a chance to be stronger than Goku even for a moment.Vegeta soon regrets this decision, but the damage has been done, and it's all due to his oversized ego.
In contrast, Goku returns from Other World on a day trip with the immense new power of Super Saiyan 3, which he gained during his training in the afterlife. Goku had no reason to do this other than just wanting to see if he could do it, a perfect example of how Goku trains for the sake of self-improvement. Vegeta was undoubtedly the furthest thing from his mind while training in the Other World, while Vegeta was definitely still obsessed with surpassing Goku even after knowing that Goku was dead.
The secret is, Vegeta doesn't just want to surpass Goku; he's jealous of Goku's ability to always break through, and he doesn't like that fact. Vegeta wants to be the one who pushes the boundaries of what's possible more than anything, and it all ultimately comes down to his identity as the Prince of All Saiyans. Can Vegeta really claim that title if he's not the strongest being out there? It's a question that's always on Vegeta's mind, and it bothers him to no end.
Will Vegeta ever get over his jealousy of Goku in Super & GT?
Of course, that begins to change in Dragon Ball Super. Vegeta, having learned from his terrible mistake with Babidi, begins to appreciate what he has built on Earth. In Battle of Gods, Vegeta protected his family and fought for something other than his ego for the first time, giving him new power.. Since training with Whis, Vegeta has learned that his mindset does indeed affect his ability to grow, and although he still obsesses over Goku to some extent, his pursuit of power is not simply to surpass Goku.
Vegeta in GT has regressed in some of these aspects compared to Super, which is unfortunate, and is more a consequence of Super being created after GT than anything else. Baby considers Vegeta's body to be the most superior body in the universe, claiming that Vegeta has the potential to surpass Goku, and further emphasizes that it is Vegeta's mindset that holds him back.
Super provides a better continuity for Vegeta's character arc, and emphasizes his growth in moments like when he teaches Cabba from Universe 6. This is exactly the kind of thing that Dragon Ball Z's version of Vegeta could never do, and it shows that Vegeta is actually learning to overcome his jealousy in some ways. Perhaps now that Vegeta is on a different path than Goku, mastering Ultra Ego instead of Ultra Instinct, Vegeta won't feel compelled to compare himself to Goku.
Vegeta still has a lot of growing up ahead of him, and hopefully he'll be able to overcome the most tragic aspect of his character since Dragon Ball Z finally done