What John Lark Wants
Mission: Impossible - Fallout has two villains working towards the same goal - to cleanse the world by putting it through hell - yet with opposite motivations. August Walker, the real John Lark, believes fully in the cause. He orchestrates Lane's release and the plutonium bombs, and in the grand scheme will be the one to lead the Apostles going forward. His goal is to achieve what the Syndicate wanted.
This is reflected in how he approaches the mission. He's introduced as the CIA hammer to Hunt's IMF knife, and while his loyalties don't lie there he is a man of brute force: a strong fighter, aggressively confident, and possessing a powerful backup. Like Lane he's also a schemer, able to repeatedly escape Ethan's grasp thanks to his advance plotting, although in arrogance repeatedly underestimates his opponents, getting discovered and repeatedly trailed when, ideally, he'd leave them behind completely; he repeatedly dismisses IMF as men in "Halloween masks" and bemoans Hunt for everything being "so f**king complicated".
What Solomon Lane Wants
Solomon Lane was already established in Mission: Impossible - Rogue Nation as a dark mirror to Ethan Hunt, committed to terror instead of safety. He was the agent gone bad, the one who willingly went rogue and took those more corruptible than Ethan. His goal in Rogue Nation was to destroy IMF and specifically its top agent. Even with his grand plan, Lane had an obsession with Hunt and desire to make him suffer; he knew he was the only one who could stop him.
In Mission: Impossible - Fallout, that returns. The Apostle plot and his attack on Ethan are intertwined, with him framing Hunt and causing a smallpox outbreak just so Walker can place Julia as a medical aid worker at the Kashmir village where the bombs will go off. Solomon Lane believes in the cause, but has personal grievances to address too. In the end, he doesn't even get a knowing death and is returned to the UK secret services.
How Ethan Hunts Wins
Lark and Lane represent two common villain archetypes: the planner and the attacker. One is obsessed with the scheme and has the hero intersect, while the other has the hero as an intrinsic part of their plot. More often than not, when the villain is intended to reflect the hero, we see these traits merged into one - see much of the MCU, Blofeld in Spectre, and Lane himself in Mission: Impossible - Rogue Nation - but Fallout delineates them into two characters, meaning the individual motivations can be greater explored and the imbalance highlighted.
After all, it's by playing on these differences that Ethan Hunt wins. Over the breakneck helicopter chase, Walker slowly loses his calm exterior to become an enraged brute attacking Hunt not because it will stop his plan (he doesn't pay much attention to the detonator at all) but out of pure emotion. His pretty boy exterior is warped by scalding engine fluid (and, finally, a giant hook) and his entire existence hinges on fighting Hunt. Conversely, Lane may have concocted the ultimate game for Ethan and spends much of the countdown toying with Benji and Ilsa, yet in the final moments becomes genuinely concerned with the plan's execution. Either villain becomes the other type when pushed against it, highlighting inherent character flaws.
That's what Mission: Impossible - Fallout's story presents, although what the movie's really focused on is reading the hero. To do that, we need to go deeper.