Superman to me is as boring as things get. His power levels change drastically based on whoever is writing, whatever was written first isn't necessarily what should be followed. There would be no plot for him at all if he was literally just like God, flying so fast around the earth that he turns back time and such.
Batman is human, and his status as a badass Everyman is the reason he is more popular. He has normal struggles, so we can relate.
A lot of writers consider Batman to be smarter, possibly in the top 5-10 of DC intelligence. He has literally had a story where he single-handedly beats the entire Justice League. He obviously cannot beat Superman in a bare hand fight, but Batman gets to play dirty while Superman never strikes first.
Basically, Batman is a better character. No matter what old stories about Nazis in the early days of Superman might say, he's really boring if he's invincible. So writers have tried to make it more canon that he's not as smart as Batman, on top of being weak to a rock , one that Batman has been shown to possess and keep on him many times.
His relevance is what you choose to believe. In the new DC Batman is a genius (as always really), and Superman gets nerfed, because writing a story for God is boring.
