Wow this thread grew fast.
Lighting Rider summoned me here, and with thunder and hel fire, I answer.
So...I feel like this is a question that would require two essays to actually get into. And for once I'm gonna try not to do that. Much. So in the very broad strokes, I figure.....
Diana's an aristocrat, and accustomed to authority and rule. In her mind, it is correct and right for the masses to trust those who have been put above them and it is natural and right for Diana to often be the one who carries that burden. However, in the philosophy of loving submission, the Amazons view authority *very* differently than we usually do; it's about submitting to the people in every respect, not influencing your own views onto others (except Man's World; the whole WW mission is to get us to accept the clearly superior Amazon philosophy).
Clark doesn't share that trust in authority, he's American. And we can see this very clearly with their "civilian" ties; Clark's a journalist, who's job is to uncover the truth and reveal corruption while Diana usually works with some from of military, government, or intelligence agency, and while she may not trust individuals in those positions, it's almost always "you betrayed your authority" and not "the system is wrong."
Having been raised to rule her entire life, Diana is likely far more selfless than Clark. Despite his legend, Clark's actually got a strong selfish streak in him (not "I'll save myself!" selfish, that's just cowardice and Clark's no coward). Hell, the entirety of the "Clark Kent" façade is a selfish act when you get right down to it. Clark will treat himself and indulge more than Diana will.
Diana's also more empathic. Clark *wants* to understand people, he tries to see things their way, but his brain simply doesn't work like our's does. He can read our DNA, but there's a psychology to being human that he'll never fully see.
Clark views the world in black and white (there is right and wrong in this universe, and that distinction is not hard to make) but Diana sees all the shades of gray, understands the variances and points the great philosophers make, she tries like hell to put herself in the shoes of other people and typically succeeds. And we see this reflected in Diana's history and all the villains she's (tried to) redeem.
Diana's also used to being the center of attention, and is comfortable in her own specialness. Diana doesn't lament that she's not a "normal" person, and wouldn't want to be. Somebody has to be the special one, the miracle baby, the divine child blessed by gods, and being special comes with responsibilities and demands and it's better for Diana to carry that weight, because it means someone else doesn't have to.
Diana knows she's special but completely misses how far above everyone else it puts her; in her mind everyone is just as special/beautiful as her, just in less obvious ways. She truly doesn't see how amazing she is, but she knows she's competent and aware of her own power (and not just muscles). With Clark, there's two distinct elements constantly pushing at him on this topic; on one hand he loves his powers, the bittersweet wonder of his lost culture, everything that sets him apart is something Clark loves about himself; it's allowed him to have such adventures, meet such people, make such a difference.
But on the other hand, he's far more aware of his Otherness than Diana, he feels his alienation more keenly. He doesn't like attention, he's actually introverted and shy (Superman's bravado is as much an act as Clark's clumsiness) but wishes he could truly fit in, while Diana assumes it's only natural to have people watching her and can mingle like a queen.
Diana's more disciplined, but Clark's more inventive. Diana's better educated in politics, social sciences, and military/strategy matters, but Clark's knowledge of physical sciences is crazy advanced. She's a far, far more skilled fighter, but Clark is harder to predict in a fight. In a sparring match, she wins 7 out of 10.
Clark will try to avoid a fight if he can, but it's easy to provoke him into action and he'll punch first and then sort out who needs more later, but he'll stop fighting as soon as the other guys says "time out." Diana is harder to provoke, puts even more effort into diplomacy, but if you make her throw a punch she's not gonna stop until every perceived threat is neutralized. Diana will kill if she must and regret the necessity but view it as one of the burdens of her mission. Clark struggles greatly with using lethal force, doesn't believe he has the authority to decide who dies, and will occasionally exile himself for such acts until he calms down.
Clark's mission is a long-term view that looks a thousand years ahead to the Legion's era, and his goal is to help us steer ourselves in that direction. Diana's mission is about the here and now, showing us that the Amazon philosophy will make us better and happier. Diana hopes to see Amazon philosophy earnestly begin it's spread across the globe within a generation, Clark expects his mission to take much longer.
And this is turning into an essay, so I'm done.