There's a concept called cognitive dissonance. Essentially it means that a person is very uncomfortable if they are faced with two confliciting beliefs. it's hardest to deal with when you're faced with something that goes against the things that are most important to you.
E.g. An honest person will hate being accused of lying. A reliable person will hate feeling they've let you down. Someone obsessed with their looks will hate it if you don't like their new haircut. It's weakest when you're faced with something you don't care about. An honest person may respect you for being honest when you tell them you hate their new haircut.
Most good villains represent the thing that the hero can't live with. they threaten the things that the hero holds most dear.
Superman / Lex isn't so much about strength vs brains as it is small town america against big business. Superman's upbringing was in a small community where people didn't have much, worked hard for what they had, and would share with each other in an instant. He grew up with a sense of community. Lex is big business. he has the ability and the money to help, but he won't. Worse, he'll grind a small community into the dust if there is something in it for him. Superman can't understand how someone can have so much and be so selfish. Lex is drawn to Superman by the fact that Lex wants power....and he sees in superman power he coverts but can't have. He loses sleep over the idea that no matter how much he has, Superman is more powerful.
The Lex / Superman dynamic was beautifully summed up in the tv show "the New Adventures of lois and Clark. Lex was explaining to people that he had the penthouse apartment in the tallest building so that everyone else had to look up to him. Superman takes that away in one line of dialogue by telling Lex that if he ever wants him he just has to look up. A truely great piece of writing.
Batman / joker is more primal. it's about order, or control might be a better word. Batman lost his parents in an alley to a random mugging. He's doing his best to force order on gotham. he wants every alley to be safe...by personally patrolling them if needs be. You get under batman's skin by making him feel out of control. You become his arch villain by killing one person in an alley, not by doing what Lex luthor does. The joker stretches this to another level. He's chaos personified. He'll crop up anywhere, do anything, kill anyone. One day he'll kill one person in an alley, the next he'll be trying to kill hundreds. He strips away the safety that batman wants to impose.
Batman / Riddler is just as interesting, but the riddler would consider batman to be his nemesis, even if it wasn't reciprocated. The Riddler is all about his intellect. He can't handle the idea that someone is smarter than him. Batman is a detective who can solve his riddles. He needs to prove he can outwit batman. In all honesty, Batman wouldn't even notice if the riddler took up white collar fraud. To outwit Batman, the riddler has to draw him in. Batman cares if you endager an innocent, so that's what the ridldler does...all to try to prove he's smarter.
In essence, your arch villain has to represent the one thing that your hero would lose sleep over. To make superman lose sleep you threaten his sense of community. to make Batman lose sleep you endanger an innocent. To make the Riddler (or Holmes or Moriarty) lose sleep you taunt them with the idea that someone else is smarter.
Craig