Superman and Batman are two of the most iconic superheroes ever created. With histories extending back into the Golden Age of Comics, their popularity stayed high and they survived generally intact for decades, a rare feat in this fickle medium. Other long-lasting DC heroes such as Green Lantern and The Flash are almost unrecognizable if you compare their first incarnation with their current portrayal, and many others have gone through extended periods of retirement before being rebooted into a more modern character.
That's not to say that they survived completely intact. Superman is infamous for picking up new superpowers at an average rate of one every five years, and early characterizations portrayed Batman as a remorseless killer with no reservations about using firearms. Their backstories are similarly convoluted... is Superman the last son of Krypton or not? Who killed Batman's parents? The answers depend on which of the dozens of versions of their histories is your favorite, and various attempts to reboot DC's continuity into a cohesive narrative have failed miserably.
Although they share similar levels of popularity and an iconic treatment by fans and writers alike, Superman and Batman represent the opposite ends of the comic book hero spectrum. This is perhaps to be expected, if they were too similar they probably wouldn't be able to share the limelight as they do. They're both squeaky-clean heroes with unassailable morals, but that's where the similarities end.
Powers and abilities
Superman has superpowers, in fact he's the prototypical superpowered comic book hero. Super strength, invulnerability, heat vision, x-ray vision, and the power of flight are just the tip of the iceberg, although in the last couple of decades attempts have been made to tone him down a bit (not too much). His greatest weakness is the rare radioactive mineral kryptonite, which saps his powers and slowly poisons him.
Batman, on the other hand, has no superpowers. He's an ordinary, mortal man, albeit one who has trained his physical and mental attributes to the absolute peak of human capability. A genius-level detective, Olympic-level athlete, and heavyweight champion master of both armed and unarmed combat; Batman could, theoretically, be killed by a single lucky shot from any 2-bit street punk with a handgun and something to prove. Some have joked that he does have one power, super-preparedness, as he always seems to have just the right gizmo in his utility belt to get him out of a jam.
But these are relatively superficial differences compared to their secret identities. Secret identities are such a staple of the genre they go largely unquestioned. On one level it makes sense, you can't be "on" all the time, and superheroes with misgivings about killing tend to accumulate a rogues gallery of vengeance-seeking criminals who would love to go after their families in retaliation. This doesn't apply to everyone, of course, for example Marvel's Fantastic Four believe they can do the world more good with public identities, and some heroes are just too weird-looking to pass for normal in a crowd.
The obvious difference in their civilian identities isn't even the important part. Sure Clark Kent is a small-town country boy, raised on a farm by All-American, salt-of-the-Earth country folks who instilled a strong sense of moral values and responsibility he carries through his entire life. And Bruce Wayne got his start as a rich kid, whose privileged life took a tragic turn at far too young an age when he saw his parents gunned down in the street, right in front of him, by a petty criminal.
But the biggest difference, at least in their modern incarnations, is that Superman is really Clark Kent, and Bruce Wayne is really Batman.
Read that sentence carefully. For Clark Kent, Superman is the mask. Superman is the costume he puts on when he has to go out and save the world. The default identity, when his mighty array of powers isn't needed, is newspaper journalist Clark Kent. Even in costume, when he relaxes his Clark Kent personality comes out. You wouldn't think this is likely, since he is after all a superpowered alien from another planet, trying hard to pretend he's human, constantly on guard against accidentally misusing his power. But he was raised that way, and likes being human. He's used to it.
For Batman, on the other hand, Bruce Wayne is the mask. His default state is Batman, prowling the streets of Gotham City by night on a never-ending mission to avenge the death of his parents and keep the streets safe for ordinary, law-abiding citizens. Bruce Wayne is the face he puts on during those unpleasant occasions when he has to interact with the mundane world, play-acting as a wealthy socialite, millionaire playboy, and business tycoon. But that's not him; he's a lonely, brooding, unhappy man still mourning the deaths of his parents, beating up super-criminals as a kind of self-punishment and therapy.
Clark Kent is Superman when the worlds needs him to be Superman. His parents raised him to be Clark Kent. Bruce Wayne is Batman because that's what his childhood made him, and Bruce Wayne is just what the world expects him to be during the day.
Addendum: TenMinJoe says There's a Batman Beyond episode where the aged Bruce Wayne is apparently hearing voices in his head, but he realizes they are a clever illusion rather than genuine psychosis because the voices call him Bruce, and "that's not what I call myself in my head"
Addendum 2: Jet-Poop reminds me that an issue of Sandman contained a shared dream populated by many of the figures in the DC universe. In this dream, Clark Kent appears as Clark Kent, and Batman as Batman.
Quentin Tarantino and Superman: "Clark Kent is how Superman views us. And what are the characteristics of Clark Kent? He's weak... He's unsure of himself... He's a coward. Clark Kent is Superman's critique on the whole human race."
— Bill from Kill Bill,
paraphrasing The Great Comic Book Heroes by Jules Feiffer (thanks Timeshredder!)
Bill (David Carradine) delivers this now-famous speech to The Bride (Uma Thurman) to try to make her see herself for what she is, she's special, she's dangerous, she's not the normal family woman she was trying to become. She doesn't have to pick up her sword to be that, like the Green Lantern's ring or Iron Man's armor, she is powerful with or without it. While dramatically appropriate to the scene, it's a relic of an obsolete interpretation of the character.
While this was certainly true, if somewhat exaggerated, of the Silver Age Superman, and his portrayal in the Christopher Reeve Superman movies, we have to remember that Bill is an older dude and probably hasn't picked up a comic book since the early 80s (Bill, if you're reading this, man, I envy you for missing the X-TREME anti-hero 90s).
The modern version of Superman is not this. Like an immigrant who came to the US as a baby and was raised in the white suburbs rather than an ethnic ghetto, he has no personal recollections of the world he came from, and although he feels a connection on some level to his cultural roots, he has never experienced them first-hand. Clark Kent's homeworld is Earth, because that's where he was raised. It's the only world he can call home. And being raised among humans means he identifies as a human being, if a special, privileged one (and even this is subject to confusion depending on whether or not, and in what way, a particular incarnation of the character adventured as Superboy in his youth or was even capable of doing so, which is wildly inconsistent).
If Superman wasn't Clark Kent, then why is he Clark Kent? The Clark persona doesn't really need to exist if it wasn't the true identity. Clark doesn't exist because Superman needs to disguise himself — he doesn't — he exists because Superman wants to have a normal life in between adventures. And because he was raised by humans, with human values, that's what he believes is a normal life. The real man wears a suit and wing-tips, not a cape and boots.
The key issue pushing the speech is that unlike other Golden Age heroes, Superman doesn't need accessories to be Superman, his powers are part of who he is. But this argument is fundamentally flawed: powers do not make one Superman, just superhuman. Bruce Wayne is clearly the more invented of the two mundane identities. Imagine what would happen to each hero if there were no more crime to fight, and no natural disasters to repair. Clark Kent would likely hang up the costume, marry Lois Lane, and never use his powers again. Bruce Wayne's life would simply lose all meaning.