First, big ups the OP, this is a legit topic, especially for hardcore DC fans (and anyone else, I guess) Probably what will always make DC stand out from Marvel is these two characters who represent so many of the different facets of the super-hero genre.
The only character that really comes close in the Marvel world is Spider-Man. But then saying that I'm a DC fan, so very biased there.
Also, this topic is deeply complicated as Batman and Superman have so many different interpretations, who's to say who is the definitive Superman (was it Reeve, Byrne's Man of Steel, the Silver Age ?) or Batman ( Miller's Dark Knight, Bale's Batman, Snyder's, O'Neill's ).
Having said all that, I can't resist having a go.
This is a tough one, and of course, just IMO.
Who would I rather be ? If I had to choose one, I'd say Superman. Sure it's a huge responsibility, and it would make one very weary, but at the end of the day Superman has the power to actually make changes in a big way, to really make a difference (sometimes that goes wrong, I met Tom Taylor when he was down here last week, and "evil" Superman is something we talked about).
Plus, at the end of a long day of saving the world, Superman can fly off into the sunset and pull himself up out of the craziness, and stand on a mountaintop.
And flying....well, who wouldn't want to be able to fly ? That's one moment I think a lot of people overlooked in Man of Steel, that the first time we hear Clark/Kal laugh is when he's learning to fly - whether Snyder intended it or not, I think he hit on possibly the greatest thing about being Superman....being able to fly.
Batman, well he's doing the best he can, but there are certainly arguments that he's as much a part of the problem, as he is the solution
(if anything that came out in the Nolan films). You have to give him full marks for trying, but if he really wanted to make a difference in Gotham city he'd use his vast wealth to reduce unemployment, set up drug treatment facilities, and improve social services (all of which would drastically reduce crime).
Plus, Batman is about buried rage, self-righteousness and revenge. He's not a particularly balanced or happy person. This of course makes him more relatable, because he's human, and there are many many unhappy, obsessive and unbalanced humans (well that's my experience so far anyway).
Don't get me wrong, I LOVE Batman, he's sheer awesomeness, but if I had the opportunity to be one of those characters, or hang out with one of them, it would be Superman.
I think what makes Superman a slightly more appealling character, is not just that he's got immense power, but what he does with that immense power - that's what makes him Super. He doesn't do what most people with power do, which is take advantage of it and subjugate and abuse those beneath him. Instead, he uses that power, selflessly to help others.
What would Batman do if he had Superman's power ? Who knows.
Regardless, these two iconic characters are the yardstick by which all other superheroes are measured. IMO Spidey comes close, but not quite.
Again this is all IMO.