Kirk's crime was worse? Only if you believe Burnham didn't do anything wrong at all. But Starfleet's view is that she assaulted a superior officer and attempted to take over the ship, with the intention of executing a "Vulcan hello." Starfleet also blames her for starting a war in which (presumably) thousands of lives and numerous ships were destroyed, which presumably stems from this: she was ordered to do "just a flyby" and to take "19 minutes, not a nanosecond longer," but exceeded that mandate by landing on the unidentified ship, resulting in an encounter with a Klingon, with disastrous consequences. Had she done a flyby and returned to report the presence of a ship, there may not have been a war. She did not defend herself at her court martial and entered a plea of guilty to the charge. The punishment for these charges and those consequences, which she did not contest, was arguably appropriate. She offered nothing in the way of mitigation of her punishment (such as, "I was right about the Vulcan hello"). She was, after further heroic acts, pardoned. (All of this, incidentally, got her off the Shenzhou and onto Discovery, which was necessary from the perspective of the Red Angel. For all we know, the Red Angel shaped the court martial with the "right" officers to ensure the sentence that was needed for the Red Angel's purposes.)
Kirk conspired with other officers to take a ship (which he recently or possibly still did captain), which included the assault and detention of other Starfleet personnel and the sabotage of another ship, and traveling to a forbidden destination against orders, with the destruction of that ship (which had already been slated for decommission) as a consequence. The only charge pursued was the refusal of a direct order. He, too, entered a plea of guilty, for which he was demoted. In way of mitigation, he saved one life (Saavik) and recovered a life previously though lost (Spock) who stood with him at the court martial. The existence of mitigating circumstances were noted at the court martial. Kirk's additional heroic act occurred prior to the court martial, which undoubtedly affect the outcome, but he was still found guilty. Burnham's subsequent heroic acts came after her sentencing, but resulted in a complete pardon.
I am not convinced that Kirk's actions were worse. They were similar except for the consequences. Kirk lost a ship that was slated for decommissioning, but actually saved lives and possibly prevented the Klingons from learning solid intel about Genesis. The
consequences of Burnham's acts are what doomed her court martial proceedings - she precipitated a costly war.
