Just to start off, I prefered the original Tim Burton Batman films over the original Superman franchise.
To define a Superhero: "A figure, especially in a comic strip or cartoon, endowed with superhuman powers and usually portrayed as fighting evil or crime."
As a superhero, Superman is flawless in terms of ability and strength, and in terms of flaws (which is resident in any superhero) is the kryptonite. This adds to the sense of out of this world power and mystery which is integral to Superman and each of the film narratives. Superman is in fact, super, in every sense of the word, from stopping a bus with one hand, right down to his flawless hollywood looks. He is an American hero embodying every aspect of what a human would love to be and be able to do.
However, Batman is a more of a vigilante, fighting crime with his pure passion, drive and strength rather than his super powers (of which he has none). Everything 'super' Batman does and has, is from the use of perfect phsyical ability, James Bond style gadgets and even sidekicks (i.e Robin and to some extent Alfred). Batman is no more a superhero than a character such as Iron Man... Manufactured super power.
Again.. However, I cast my vote for Batman in terms of who is the greatest superhero. As Batman does embody some traits of which the definition states above, he cannot be ruled out as a superhero, he is in fact one of the greatest in terms of how he fights crime on a level almost similar to what we see in todays society (although in the films, the context and portrayal is somewhat exaggerated and 'hollywoodesque'). Therefore Batman is a character of which most cities would love to exist...a supercop.
Batman does win on terms of 'realism' which as was mentioned above is a kind of shallow argument.. Sooo, maybe on terms of how the audience relates to the superhero, Batman is more approachable, he has real problems which could be faced in real life, whereas Superman's weakness is materialistic and above all 'fake'.