While I really like and enjoy both characters immensely, I also prefer Superman hands down. Grew up constantly watching the Christopher Reeve films. While I do think that traditionally Superman is DC's marquee character (and marquee superhero in any universe, as I feel Supes is to superheroes what the Rose Bowl is to college bowl games, "The Granddaddy of Them All") I think there's no denying that Batman is currently DC's most popular character.
I think a lot of that stems from the fact that many people these days seem to like their heroes "flawed". The same way many people may find Iron Man/Tony Stark appealing on the Marvel side. When characters that are otherwise intended to be "above" humanity suffer from the same everyday flaws that we do in everyday life, it makes them more relatable to us. It also I feel makes us feel as if these heroes aren't so "above" us after all. This is why I think both recent Superman and Captain America films (TWS and CW at least) are a bit more dark and serious in tone, because they are both dealing with characters that are meant to represent something that absolute and incorruptable without having a dark past; in this case, absolute "good". The notion that something is so pure I think is something that many people struggle with, because many people don't feel this is relatable and/or it potentially reminds them of what they aren't. Similar to the way Tony Stark despises Steve Rogers.
What they are currently doing with Supes I feel is the only way to make him relatable while still maintaining his aura of goodness at the same time. He can't be related to on a physical level (the way Username mentioned Batman can be) because he's literally a God like being, so he has to be made to feel and struggle with the same type of emotions that we as everyday people do. That's actually his biggest vulnerability that has never been tapped into nor explored yet, but is being done so in this current DCEU. How he's depicted to handle them though is a testament to his character. Being forced to confront your ideals and values I don't feel is an abandonment of them; it's a reinforcement of them by testing how steadfast you are to believing in them. In the DCEU, as opposed to making him be an alien that is trying to fit in with mankind, it feels they're making him out to be an everyday person who is struggling with the fact that he is an alien and possesses these superior abilities, and is trying to do what he can with them to help. Playing the hand you were dealt, so to speak, and trying to make the best of it.
While it is good to feel a relation to your heroes (being that's part of the reason that you feel drawn to a particular character in the first place), I also at the same time think anyone you idolize is indeed someone who you should feel is better than you. Someone you look up to shouldn't be someone you want to be able see eye to eye with and feel that you're just like them, because then there's no reason to idolize them. That's why they call it "looking up" to someone or something. They inspire you because you aspire to be more like them.