The best and the brightest—or darkest

Batman- Captain America

This one is very problematic…but I can’t quite let go of it…

Here we have the two characters that represent the upper limit of human capability —meaning capabilities unenhanced by extraordinary means. Of course Steve Rogers was enhanced by “vita rays,” but he was always played as having achieved the pinnacle of what one man can do unassisted by means common in comic books.

As has Bruce Wayne. Since 1986’s Dark Knight, however, Batman has been characterized as a dark, obsessive and, frankly, not a very pleasant individual. Whereas Rogers has always been written as a sunny, ineffably decent man specific to an idealized “greatest generation” (the “Ultimate” iteration differs, in that Rogers is a hard-bitten, quite violent soldier).

In the last couple of years, writer Grant Morrison has evidently reverted the Batman character to the somewhat dark but not at all psychopathic iteration known from Dennis O’Neil and Neal Adams run in the early 1970s. So it is this version paired best with Cap—certainly not Frank Miller’s “Goddamn” version. In any case, in both 1995's DC vs Marvel event and 2003's JLA/Avengers series, both were seen as each other's equivalent.

Commonalities:
The above said “upper limit of human capabilities.” Both are respected in their respective diasporas for that quality and for their unparalleled tactical cunning.

Differences:
Wayne is one of the richest men in the DCU with deep roots in Gotham society, and is motivated to avenge his parents murder. Rogers is motivated purely to fight for right and the American Way. He has also often been depicted without much of a personal life, a perpetual “fish out of water” coming to terms with living in a world so different from the World War II era. In pure rich guy terms, “Bruce Wayne” = “Tony Stark” would be much more apt. In pure "dark avenger of the night" terms, Daredevil might be better…but whatever…

Alternate histories:

BM: Bruce Wayne is born in the 1920s and receives special training as a young adult to work as the U.S.’s most feared operative overseas against the Axis Powers. He is joined in these efforts by orphan Dick Grayson and allies like Aquaman, until he is thrown into suspended animation. He is revived by the Justice League of America, which he joins as its longest-running leader and is universally respected as the most inspiring in the super-hero community. Wayne partners with Black Lightning for a time and struggles to forge a personal life in a time much different from the one in which he was born. After he leads opposition to an effort to register superhuman and costumed operatives, he is assassinated.

CA: Child of privilege Steve Rogers trains himself to be the defender of both the metropolis of his birth and the nation at large and the ultimate opponent of terrorists and superhuman criminals extant. He mentors a number of younger operatives, works closely with police and the military and helps found the Avengers.