Strength
Batman: 1000 lbs, possibly a bit more
Captain America: At his lowest, 1200, but in practice 5 tons at least, at most? 10 tons. He's been displayed easily overpowering people on this level, and has quite a few feats that could put him at 100 ton level, but I don't use PIS, which is what those were, but keep in mind they exist
So this puts Cap easily above Batman in strength, and at least should be able to overpower him, but let's keep going.
Speed
Batman: 26-30 MPH
Captain America: 50-60 MPH
This allows Cap to easily close the distance with Batman, bringing the fight into close quarters where Cap doesn't have to deal with many of Batman's toys.
Agility/Reflexes
Batman: 100 miliseconds maybe 80 miliseconds since this is comics, but no more.
Captain America: 4-5 miliseconds, At least 16x faster than Batman, at most? 25x faster. That means should Batman strike him, Cap's reflexes would allow him to dodge and retaliate much faster than Batman could react.
Durability
There are no numbers for this, but Cap would take it for the pure fact that, he takes more beatings than Batman, he survives larger-scale explosions, and tends to fight enemies like Iron Man-a 100+ tonner-and comes out fine.
Stamina
Captain America by far, Batman is human, he can get tired, Cap can't. If this turned into a war of attrition, Cap would take it.
Martial Prowess
Wash, both are masters of every single Martial Art, but if I had to, I'd give it more to Cap due to his experience with Asgardian, Olympian, Wakandan, and K'un-Lun fighting styles, though I don't know if Wonder Woman taught him Amazonian Martial Arts, or Superman Kryptonian, so it could easily wash once more.
Now, with pure strength Cap likely couldn't win against someone of Batman's skill, but with equal skill, that causes Cap to shoot up because he can then use the same techniques as Batman, plus more possibly, and he then has the strength to put out more damage, the reflexes to dodge more attacks, the durability to take more hits, and the stamina to keep hitting. That's what matters when two people are equal in skill on this level.
But then you also need to remember, Cap has a close-range weapon, which when it comes to Martial Arts, automatically gives someone a lethal advantage, sending Cap into the lead.
Defensive/Offensive Equipment
Now with armor alone I'd wash this out again, as both are made of lightweight Titanium and can withstand massive punishment, but you also have to take Cap's shield into this since that sounds as defense, and as a Proto-Adamantium/Vibranium shield, with Uru to enhance it as well, Batman does not have one weapon that can pierce that shield in his arsenal. While it may not cover enough of his body, consider this.
Both of their armors themselves might be equal, but their weapons determine who will bust through what, Batman's weapons won't go through Cap's armor easily, it's fire retardant, electrical retardant, etc.... but Cap's shield, that's gone through Adamantium once before, and regularly goes through Titanium armors, including that of Iron Man-who's even complimented Cap on the durability of his armor before. Which means should both of their weapons impact the other's armor, Cap's armor will hold up, Batman's won't
Versatility
As much as Cap can use his shield for, and all he can do with it, he can't match Batman's versatility, not by a longshot, Batman can do too much with his many weapons and that gives him the win, and being able to throw multiple weapons at once can truly do some damage.
Logistics
Cap, and while you may question this one, Cap's ability to reuse his shield over and over again means that he's not running out of ammo any time soon, giving Cap the advantage in this match. Eventually, no matter how long it takes, Batman's going to run out of ammo.
Batman's stealth
Now this category isn't just called 'Stealth', and you'll see why.
Batman is the master of stealth, very few have his abilities, he's a ninja and almost nobody can sense him, but nobody doesn't include Captain America.
The thing about Batman that people use is non-combat feats, sneaking around Superman when he's not paying attention, sneaking around other Superhumans when they're distracted with others and they can't focus on him. Besides, hearing his heartbeat doesn't work because not only does his armor quiet his heartbeat, but he has methods to slow it down, but Cap won't use that. Cap uses air currents and movement, and as someone with mass, Batman is subject to these. As he moves around, he interrupts natural air flow, and since two forces cannot occupy the same space at the same time, as Batman moves around he pushes around the air, and Cap can feel this disruption in currents, or the increase in force as Batman moves towards him, and can react to that, even in a smoke bomb he can do that and feel the movement and react. Not only that, but his eyes are enhanced to the point he'd sometimes been able to see perfectly in the dark with only the slightest bit of light around him. Even if both of those failed, this is a combat scenario and it's much harder not to make noise in combat and Cap could use that, easily. And then to really top it off, Cap's reflexes would make it entirely possible for him to simply grab Batman after the first strike and then wail on him, hard.
Strategy
Probably the biggest surprise to most people, Cap is the better strategist, as good a strategist as Batman is, as much as he can do with his mind, as much as he's done and he's worked for, it comes down to he's only human. It takes him time to come up with his best strategies. His strategies for the Justice League took weeks or months of prep and he tends to focus on an enemy's weak spot or something he can exploit, Cap doesn't have that when it comes down to it, at least not easily detectable. Cap's different, his strategies are more grounded in simply working the battlefield to his favor, fighting until there's an opening and taking it, you have Batman taking on Superman, Cap taking on Hulk, King Thor, Iron Man, Cap doesn't use their weaknesses against them, he simply fights, and wins with his strategy. He doesn't need to work it out beforehand.
Then like I said, Batman takes too long, he needs to develop something to win, he needs to develop a counter-measure, and he needs time to do it. I think a guy I met once put it nicely. If you tell Batman and Cap to kill a man who was going to conquer the entire world in a month-and their morals were removed-they'd go about it differently. Giving them the same info, same intel, they'd both come up with different results. Batman would analyze the data, spend about a week coming up with this analysis and then once he'd analyzed their every move, come up with every extra piece of info he could, he'd then spend another week working out the exact equipment he'd use, and then another working out his exact movements, and only then he'd move, and he'd do it without so much as a single witness.
Cap will calculate everything that night, and have a body on your desk by the next morning, 5 o'clock at the latest. That's the difference in their strategies, Cap simply calculates things faster, thinks them through faster, and he doesn't have so much tech to sift through, he just does it, and while more people might see him, he gets the job done faster and just as good, he just goes about the actual job differently.
In the end, as good as Batman is, his only real advantage is his versatility, and Cap's durability as well as his armor and shield work for him perfectly in that aspect.
The winner is Captain America.
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