I love the odd bits of Superman/Batman parallelism in terms of, well, everything. Even though people (and by people, I mean comic book writers) only seem to be able to compare their origins, as if that's the only moments of true parallelism.
I love the idea that Superman's personas are both inherently true, as I believe that Clark is as much him as Superman is. I love the idea that neither of Batman's personas are true to him, like, at all. Bruce Wayne is, well, Bruce Wayne, idiotic playboy bazillionaire. Batman is sort of this act that he puts on to scare criminals. Yeah, it's more true than Bruce Wayne is, but I still think it's very much an act, while Superman is both badass superhero and dorky reporter (which is why it's such a big deal for him to need Lois to fall in love with Clark as well as Superman, because they're both him, and he needs her to love all of him, not just a part).
I love the fact that both Metropolis and Gotham are New York City, except that Metropolis is utopia!NYC and Gotham is dystopia!NYC. I have this whole thing about cities, as you may or may not have figured out, and the idea that they come from basically the same exact city, except completely different. I love the idea that their cities have shaped them, in some way, or that they've somehow shaped their cities.
I love that Batman is from this planet, yet alienated enough that he has to create a persona to deal with it. I love that Superman is from another planet, yet loves this one enough to live among the people, creating a persona in order to do good in it.
I love that Batman goes underground, creates a cave for himself to watch the goings on in Gotham. I love that Superman goes above the world, likes to watch it from the outer atmosphere (because, uh, there's no sound in space).
I love that Batman chooses to use fear, and that Superman chooses to use hope. That they chose those things, because they believe it's the right thing to do, and I love that sometimes they're right, and sometimes they're wrong.
I love that Batman is so rooted in one place, despite having been around the world. His town is Gotham, and that's where he works, that's where he does business, that's what he loves. I love that Superman is about the entire world, even though he grew up in a small town. He loves the world, he protects all of it, not just Metropolis.
I love that they're big damn heroes (to borrow a phrase from Joss Whedon), that they both wear silly costumes, that and that they're always there to save the day.
P.S. I'm totally this poster to go along with this one. Because they seriously match. I swear.