Better Costume: Batman
Better Martial-artist: Batman
Stronger: Cap
Faster: Cap
Smarter: Batman
More badass (give example): Batman. Kryptonite slaps Superman.
Better background: Batman
More interesting character: Batman
Cap still beats his ass in a fight, but Batman remains the cooler. Forever and ever amen.
Batman is way cooler than any version of Cpt America - however I do think Cpt America is superior in the physical department than Batman, however in the strategy department and even stealth (yes one of Batman's MAIN attributes IMO) is where Batman leads the way, and is perhaps somthing to think about.
In my opinion
Better Costume: Cap (Balaclava hood is cool but not the helmet he wears..........)
Better Martial-artist: Probably Bats
Stronger: Cap
Faster: Cap
Smarter: Probably Bats
More badass (give example): Cap (manipulated Hulk into killing Nazi alien)
Better background: Cap (He has the whole world war, got frozen in ice but i dont care attitude)
More interesting character: Cap