Superman is am more popular symbol and icon, his iconography is used far more in random conversation, but Batman is a more popular character. that being said, Batman seems to appeal to people more in modern times. I think it's because parts of Superman's mythos need serious updating. Example, newspapers are becoming more and more irrelevant by the year, and don't have the symbolic significance they once did. But making Superman a blogger or newscaster kind of doesn't work, because they lack the symbolism that newspaper reporters use to. The values of middle America have also come into question in recent years. A major part of Superman's lore tells us that being raised in middle America is what made him what he is. What happens to a character based on certain conventions, when those conventions are perceived as problematic?
Batman, has also had his basic premise questioned in recent years, but he's been able to weather it better then Superman, because he is a more versatile character. A good example of this was Snyder's Superman film. A sizable percentage of Superman fans hate that film with a passion because they don't feel it represents the character's essence well. Now look at the Batman films. You've had campy Batman, films (Adam West) Gothic Batman films (Burton) "realistic," Batman films ( Nolan) and even goofy family film Batman(Lego), and you rarely here people say that any of these versions "don't get the character". Their all just versions of Batman. I think this stems from the fact that what Batman represents is more nebulous then what Superman represents, and can be morphed to fit the times a lot more easily. It's hard to change Superman significantly without loosing what makes Superman "Superman". What if Kal-El's ship had crashed in Compton rather than Kansas? would his values be different? Or would it be the same character?
Batman, at it's core is a concept that has been remade several times since the 1700's. Batman, the Shadow , Zorro, and the Scarlet Pimpernel, are essentially the same character just updated slighty for the time. But the values that make Superman what he is stems pretty firmly from mid 20th century American values and notions.