Ordinary night, ordinary rooftop. Could be anywhere. New York, Gotham, Metropolis - but let's just go with anywhere so we don't have other people involved. We can waive any parts about Batman intimidation or Spidey talking as any means of getting to the other. We know that this rarely works on either of them, as is. Let's just say that they're there, these buildings are their playground, and continue.
If we're on the basis of a flatfooted encounter with no prior encounter of each other, we have to still give Bats the better head on his shoulders, who - in his repertoire - will have seen many other opponents in his time, and apart from what his detective-level observational skills tell him, he's got information in his head that can lead him to some fairly-quick assessments without ever having heard of the Amazing Spider-Man. I had a brief look. Doesn't appear to be any like Spider-Man, but all the villains he's put down and justice people he's encountered would lead him to deduce - on sight - that he is acrobatic and physical, but his body frame and voice suggests too young to have received any formal training.
So, the fight begins, and you know what they're doing for openers. Spider-Man is using webs as soon as Batman throws anything, uses super agility to get around, his spider-sense to help him avoid getting punched so he can fight this guy. We know what they're doing in their heads as they attempt to beat each other. They're trying to get a good feel for what the other can do and react to that, find the weakness and use it. The problem here is that Batman is an encyclopedia and Spidey is a laundry list. They're separated by a few physical attributes and a sense of danger that pings when something is wrong. The problem with Batman is that he's always a threat, and Spidey has to be able to interpret WHAT is going to happen. He's not psychic. He cannot predict what will happen in a sleight-of-hand gesture, say, only that it happened.
Now, in the short time that they are testing each other out, gadget to webs, fighting style VS improvised style, Bats has learned that he's fighting a person who has enhanced physical prowess and webs, and he can't help but notice that even when he uses "The darkness as his ally", Spidey seems to have a good sense for his location. Not a perfect idea, but enough to not be surprised by a stealth attack because of his senses and reaction time. It's allowed him to even blind-fight well, despite not having Bruce's incredible training. This IS a worthy match, a fight to be remembered. The problem is that we know that Batman also tends to change tactics on his foes rather swiftly, make everything an uphill battle for them when he's at the disadvantage. He makes them exert more effort, work harder while he does less in a controlled manner.
Spider-Man attacks and then, there is smoke, a thick shroud from Batman's smoke bombs. He's got to be kidding, right? His Spider-Senses will be working on overdrive. Spider-Man decides to just hop and web out of there- Web gets cut mid-air. Sounded like a couple of batarangs. A grappler shoots at his feet, but there's a danger sense, so he leaps immediately into the air to avoid the grappler with the explosive head. Spidey webs into the smoke, which he's above now, trying to catch Batman. It's not working. He can sense him in there, but he can't pinpoint him, and not only can Batman judge by listening to the incoming webs, he has heat vision and can see Spider-Man pointing with his hands, so that even if the webs were no special temperature, he can avoid them.
The smoke clears. There's web-shot and impact webbing and what-have-you all over the place, but Bats is nowhere near it. He's standing there in the typical Batman pose of the cape surrounding his body, for the moment. He's drawn this out, but it's clear he's not going to any longer. The fight renews, ducking and avoiding each other's blows, punches and kicks traded, and though Spider-Man is physically stronger, Batman is turning the blows with his body to make them glances when he can't avoid them. It means they only hurt. Eventually, Spider-Man DOES land a good solid kick, sending Bats tumbling hard. And then, his leg explodes.
During the course of the fight, Batman observed not only Spider-Man's abilities, but his equipment, specifically his webbing, an ingenius directable adhesive whose tensile qualities - once hardened - are incredible. In the smoke and probably when he might've been caught, he determined that it could be cut, with effort and with a fine blade like his own equipment. Getting Spidey to supply him any was a cinch. He just needed to get him to act. He'd never risk himself in the mist, so the webbing was used...and cut while pressing a grenade against it. Once the fog was lifted, he got back into the fight, except that it was mostly defensive, keeping him at bay with martial arts before making it look like he made a mistake and took a solid hit to plant the grenade on Spider-Man's leg. Spidey has the web-dissolving fluid, but he can't apply it in two seconds directly after kicking a man hard enough to send him flying.
He gets up to find that Spider-Man is either fatally wounded or already dead. Those things can wreck a humanoid robot, you know. It was a good thing he planned to take that hit or he'd be dead as well. As it stands, though, he was going to need alot of recovery after this. All his training, all his knowledge, and it was the samurai-like nature remarked upon him in Japan that allowed Spider-Man to do more of the work - to even take wounds in order to grasp victory - and his magician's sleight-of-hand.
