There's no way Batman cuts out the crime in NYC. Batman isn't used to running around a city that is full of a majority of the supers, both hero and villain. These people in NYC aren't gonna give a shit some Daredevil wannabe (In their eyes) is taking the place of Spider-Man. Heck, that's just going to make them all go nuts and try shit. Not that Batman wouldn't be able to handle himself, but I don't see him making any large difference and Spidey's Rogue Gallery pretty much all being supers would pose a problem, that he would inevitably find a solution for.
Spidey, on the other hand, also wouldn't make a dent in Gotham's crime either. As many have said, Gotham is basically fucking cursed. I don't care he has superpowers, he's going to treat Batman's Rogue Gallery with kid gloves like he does any other non-super. From a purely physical standpoint, he could certainly take them out much quicker than Batman, but I don't see it really making a difference. Most of Batman's villains aren't necessarily problems from a physical standpoint like Spider-Man's are. They're psychological, but the thing is, Spidey deals with that too, so he would also find a solution.
Assuming we're talking about Spidey and Bats that both have years of experience under their belts and aren't just starting out as well.
