Sadly, Vegeta can never surpass Goku. However, both of them have their strengths and weaknesses like that of any fighter. The two Saiyans have always had a sense of rivalry, and putting their rough start aside, have become close friends as well as sparring partners.
Both have had different training techniques and even though at times, Vegeta surpasses Goku in terms of power level, in a shorter amount of time, Goku easily excels and Vegeta is left in the dust yet again. Power level aside, it comes down to the attitude of each fighter. Goku yearns to become stronger just because he can, and so he can challenge stronger opponents. On the other hand, Vegeta continues to train, in theory, in hopes to someday surpass Goku and maintain his pride. Here is the analysis of what Goku can do that Vegeta can't, and vise versa.
10 Goku Can: Ultra Instinct
During the Tournament of Power, Goku was able to unlock the Ultra Instinct form, as well as the completed Autonomous Ultra Instinct form by breaking his limits and tapping into his hidden strength while battling some difficult enemies. In this form, Goku's body constantly emanates a blue, sparkling aura that creeps upwards and he takes on a more stern, relaxed appearance. The ground around him also breaks and shatters due to his immense God-like power and energy. Within the Autonomous form, Goku can subconsciously make instantaneous reactions to the opponents' movements as well as execute his own, which makes his offense, as well as defense, greatly enhanced. In this form, some of Goku's signature moves, such as Kame-hame-ha and Spirit Bomb are elevated.
9 Vegeta Can: Kill Easily
No stranger to ruthlessness, many have fallen at the hand of Vegeta, the Prince of all Saiyans. Even after he mellows out from joining with Goku and the other Z Fighters, getting married and starting a family, he still has this undeniable streak due to his warrior upbringing. This is something that Goku has not been able to do since Dragon Ball, possibly due to his training with Kami, but also, since if he kills all the strong enemies, there will be no one to fight. One of the first to die with little remorse was his subordinate and fellow Saiyan warrior, Nappa, using his Galaxy Breaker attack to practically explode him from existence after losing in a fight with Goku. A few more to note, were one of Frieza's henchmen, Cui, countless Nakemians (including women and children), Pretty boy Zarbon, Android 19, and many civilians while in Majin form under Babidi's control.
8 Goku Can: Kaioken/Kame-hame-ha
As a young boy, Goku trained under Master Roshi along with his best friend, Krillin. This training consisted of martial arts along with signature moves created by the Master. The signature attack of the Kame School under Master Roshi, is the Kame-hame-ha Wave, which is a concentrated energy blast emitted from the hands using one's ki. The move was perfected by the Master after fifty years of training and is a move that is used the most throughout the series by Goku.
Also, the Kaioken, a move of King Kai's invention, is one that only Goku can execute successfully. This move is one that multiplies one's ki and thus increasing their speed and power immensely the more it is used. However, this move takes a heavy toll on the body and should be used sparingly.
7 Vegeta Can: Super Saiyan Blue Evolution
This power can be compared to Goku's Ultra Instinct form in that it is one that can evolve once the users' limitations are breached and their power levels are almost on par. Vegeta's Super Saiyan Blue Evolution or Evolved is a power that he achieves during the Tournament of Power, and is beyond that of Blue, even though it looks rather similar. One key differential is the appearance of his eyes, which in this form, is more of a royal blue and the pupils appear a bit large in contrast. Also, Vegeta's body gains more muscle mass. Again, similar to Goku's Ultra Instinct, this evolution is achieved by him wanting to beat a stronger enemy than himself and breaking through his limitations to achieve that. Also, Vegeta's speed, stamina, and agility was heightened.
6 Goku Can: Super Saiyan 3
The third form of Super Saiyan, and one that many other Saiyans have been able to accomplish, including Goku, Trunks, and Goten, is that of Super Saiyan 3. Goku was the first to achieve this form due to his rigorous training in Other World. It has been theorized that Vegeta has not been able to achieve his form, or possibly, has had no interest since, with the bulk of the form, speed is greatly diminished. It has also been theorized that Vegeta thinks is laughable and has willingly skilled it all together straight to Blue (or in the non-canon Dragon Ball GT, Super Saiyan 4, to be discussed below).
5 Vegeta Can: Super Saiyan 4 (With Help)
Making its first appearance in the un-canon Dragon Ball GT, Super Saiyan 4 is a form apart from the usual strain of Super Saiyan. Goku can achieve this, however, not in the way that Vegeta does. This was made possible to Vegeta due to some help from the Capsule Corporation's 'Blutz Wave Generator' created by Bulma while under the control of Baby. With the generator, Vegeta can transform into the Golden Great Ape form and then to Super Saiyan 4, to Goku's amazement.
4 Goku Can: Instant Transmission
After crash-landing on a distant planet after his fight with Frieza and his henchmen on Namek, Goku is befriended by an alien race who shares with him a very handy move. While training on this Planet Yardrat, he is taught a teleportation move made possible through deep concentration called Instant Transmission.
Goku can also teleport anywhere to anyone in the Universe as long as he can sense the person's ki. One can also bring someone else along for the ride by a touch of their shoulder.
3 Vegeta Can: Rage Mode
Due to Vegeta being able to control this enraged, or berserker mode, when he saw his wife, Bulma, harmed by Beers, the God of Destruction, he used the immense distress and anger he felt to harness a new power. This rage mode transforms his body as well as quickly elevates his power level, so much so, that he can hold his own against Beerus for a brief period. Other characters that have used rage mode throughout the series have been Frieza, Beerus and Broly.
2 Goku Can: Spirit Bomb
While training with King Kai in the Other World, Goku is taught the Spirit Bomb technique. This is available to use once a large amount of energy from the surrounding area is harnessed into one concentrated sphere. This being one of Goku's signature moves, is not one that is used often, being that has such massive destructive power and could destroy an entire planet if it is misfired. This is usually used as a 'last resort' but there is a chance it can be deflected, as demonstrated by Kid Buu, but, it can also be deflected by one who possesses a pure heart. Some of the other characters that can use this technique are Future Trunks, Krillin, Cell and Goku Black/Zamasu.
1 Vegeta Can: Defeat A God Without Ultra Instinct
During the Tournament of Power, Vegeta matches up against Toppo, who had cast his sense of justice aside and transcended to that of a God of Destruction. To defeat him, Vegeta had to use his powers in new ways, since every attack he would throw at Toppo who be absorbed or deflected. As a result, the Saiyan had to concentrate his ki while in Super Saiyan Blue Evolution mode to beat him, which was a risky move similar to that of when he practically self-destructed when fighting Buu. Not as powerful as Goku's Ultra Instinct, his attack was enough to deflect Toppo's destructive ki blast and knock him out.