Who would win in a fight:
Batman's the clear winner, and here's why
Batman could roughly match Cap's speed, if not his strength. However he also has a shitload of gadgets including smoke bombs, batarangs, and that rope-thingy. What does that mean? It means he can distract Cap enough with his gadgets and sheer cunning (arguably Batman's greatest asset) to get his shield away from him, and because Batman is an excellent strategist he'll see this as the obvious first action to make. Take the shield away it'll be a fight between the two WITHOUT Cap's shield, which will put him at a huge handicap since he's used to a fighting technique based around it, and Batman knows so many different martial arts it's insane AAAAAAAAAAAAND he still has a shitload of gadgets, can't forget those.
Now, let's look at how the two will fare in a fight without Cap's shield. I don't have any statistics or anything, but the super soldier serum boosted Cap's speed and strength, leaving him a formidable opponent even without a shield. Batman doesn't have the SSS, but was trained extremely well, and also possesses great strength and speed. I would say that it can't rival Cap's, but according to Winter Soldier Cap almost got his ass whooped by some two-bit boat hijacker. Replace that hijacker with a man who's been extensively trained AND has protective armor and several gadgets at his disposal as well as the ability to actively go in and out of the cover of darkness during a fight.
Now, until Batman and Cap actually fight we won't know for sure, but my money's on Batman, for obvious reasons.
Which one do I consider better:
That one's much tougher for me to decide. I know Batsy would win in a fight, but if that was the only qualifier then Superman would be my favorite every time with no contest. In the end, I like Cap's backstory and natural abilities a lot more than Batman's. Don't get me wrong, I love Batman, but Captain America just holds a slightly higher place in my favoirtes.