WARNING: The following contains spoilers from Chapter #71 of the Dragon Ball Super manga, written by Akira Toriyama and illustrated by Toyotarou.
Goku's and Vegeta's famous rivalry continues to drive them further into the gods' domain during Dragon Ball Super. As Goku has been training to master Ultra Instinct with Whis, Vegeta practices Destruction under Lord Beerus. The God of Destruction and his Angel attendant are now in competition to see which of their students' new techniques is more powerful.
Despite being wielded by very different beings, Ultra Instinct and Destruction are actually quite similar in that they're more of a state of being than a power-up. Both require the user's mind to be free of all outside thoughts and emotions. In doing so, the user achieves an almost limitless source of power. But which of the two is stronger?
Goku's Ultra Instinct
Ultra Instinct is a technique used by the Angels, allowing the body to move on its own without the user needing to think. This state allows the user to move faster, avoid attacks, and strike with more decisive blows. Achieving the ability requires a calm, tranquil heart -- which is why it's more suited to Goku than Vegeta.
Whis, who is always in the Ultra Instinct state, demonstrates that Goku is actually quite weak compared to the Angels. Goku fails to bring out the full potential of Ultra Instinct because he thinks of it as a transformation. Therefore, Whis tasks him with mastering Ultra Instinct in his base form. This way, he will be spared from the intense stamina drain of classic Super Saiyan transformations. Whis recommends that Goku's white-haired form be reserved as a last resort when the Saiyan must surpass his previous limits.
Destruction is a devastating technique used by the aptly named Gods of Destruction. Blowing things up is old news in Dragon Ball Super. However, Destruction is a different beast. It has the power to cause something or someone to cease to exist. The actual destruction of matter itself. As Lord Beerus puts it in Chapter 70 of the manga, "turn something into nothing." He demonstrates that Destruction can be used with extreme precision by destroying a small portion of Vegeta's armor.
Beerus explains that while Destruction does not require a pure heart, it does require a clear mind. At his teacher's prompting, Vegeta reveals that he still carries the weight of the Saiyan race's sins. When goaded into attacking Beerus, the Saiyan Prince is easily defeated. Beerus uses this to teach him that his mind must be clear of all thoughts but one: Destruction. Vegeta must let go of the past if he intends to master Destruction.
Ultra Instinct Vs. Destruction
The pattern of Goku achieving a new form and Vegeta riding his coattails is finally over -- a separate path to power for the Saiyan Prince is a much-needed change in the Dragon Ball story. Their rivalry has become the very center of the series. As such, there must be cause for a legitimate argument as to who is stronger. Goku and Vegeta have become proxies for Whis and Beerus, while the latter have become proxies for the audience. They all have the same questions. Ultra Instinct or Destruction? Goku or Vegeta?
The normally lethargic Beerus has taken Vegeta as his pupil to prove that Destruction is the superior technique. Whis continues to train Goku in the ways of Ultra Instinct, appearing to accept Beerus's challenge. Dragon Ball Super has spent much more time on Ultra Instinct, so the full extent of Destruction's power is not yet known. Neither Whis nor Beerus have ever been shown exerting even a marginal amount of effort. However, Whis is able to easily knock Beerus unconscious. This is about the only evidence given so far that Ultra Instinct could be more powerful than Destruction.