Sad to say, there seems to be an unfortunate amount of ignorance on this matter, so Ill be forced to set you all straight with a rather longer argument than I had initially intended. :::clears throat:::
Captain America has little chance of winning a fight with Beast. To define my terms, Captain America is Steve Rogers, with the Super Soldier Serum and with his trademark adamantium-vibranium shield. "Little chance" means that he would win less than half the time. A fight means a one-on-one battle, in a neutral environment (Lets say, a 30X30X30 cube with no doors or windows. Sufficient for both Hanks agility and Caps shield) without any other objects lying about that can be used as weapons. Both combatants are fighting to the best of their ability (i.e., no mind control, no pre-existing anger or tension, and no regard to their past friendship). Neither has any planning time prior to the beginning of the battle. Winning is either immobilizing the opponent to the point where they would not be able to free themselves within, say, an hour, without outside help, or knocking them unconscious (Killing isnt likely to be a factor in the fight, though that can also be considered a win). Beast is Hank McCoy, in his post-X-Factor era. Now that thats all settled
Captain America is renowned for his fighting spirit, his stance as the worlds foremost hand to hand combatant, and his cunning strategic mind. His shield is an indestructible, and unique, weapon and defense. He is trained to the height of every human capability.
Beast is, as are all mutants, of an incredibly strong will, and has trained for ten years to be an excellent hand to hand combatant, with a fairly good strategic mind. His every attribute is far beyond human capability.
From this simple a summary, results are inconclusive at best. A more direct comparison of attributes is therefor necessary. Ill give this six distinctive categories, strength, speed, agility, durability, fighting skill, and fighting spirit. Afterwards, Ill discuss the shield, and give a quick run down of how I think a battle like this would go.
Strength: There is simply no comparison here. Steve Rogers at his best can lift 800 pounds over his head. Beast can lift five subway cars over his head without breaking a sweat. Id wager that Hank is about 100 times strong as Cap. Easily strong enough to do serious damage with every blow, but, more importantly, Cap isnt going to be able to effectively block his shots with anything but his shield. A big losing situation for Cap.
Speed: A closer situation, but still far removed. Cap can run at 25 mph, Beast can hit 110. Cap can dodge missiles, Beast can catch them. Cap can throw his shield at 120 mph, Beast can throw a baseball at mach2. This is gravely unfortunate for Steve. Hes fast, and hes an incredible fighter, but Beast will see every move, and have plenty of time to stop it. Even if Caps already moving, Hank still has the fraction of a second he needs to block. Even sans shield, Cap can bruise Hank, but only if he connects. He wont. Meanwhile, Hanks blows will be extremely difficult to block or dodge, and if Hank uses multiple attacks, Caps not fast enough to block them all.
Agility: Similar to speed, Hank outclasses Cap all the way here. Caps agile, he can do handstands and flips, but he cant do the same kind of complicated mid-air maneuvering Hank can, and he cant use his hands and feet as well as Hank can. Hank can keep one hand on the ground, attack high with one foot, attack low with his other hand, and keep his other foot free to block an attack. Cap cant do that, and hes lucky if he can avoid either attack, given the speed they come at. Also, Caps throws and other maneuvers wont be much good when Hank can put out a hand and stop himself with greater speed than Cap could ever muster, or flip in the air with enough force to drag Cap with him, and land right on Steves back. Cap has the skill to make the right choices in battle, Hanks got the abilities to have more choices, and to always have time to execute them before his face hits the floor.
Durability: Caps tough. Hes as tough as a human can be. He wears chain mail body armor. He can take bullets and keep coming. Caps tough. Beast is tougher. His body is denser than a normal humans, he has fur to cushion him, he has muscles a foot thick to protect his skeleton and vitals, and hes got a mutant metabolism that helps him heal injuries right quick. Cap can hurt him, but its gonna take a lot of punishment to put him down, and very few of Caps attacks are going to throw Hank off-balance enough for a follow up. Hank, on the other hand, could seriously injure Cap with a single shot. If he leaped on his back the way he flattened Delgado, Cap would probably be killed. Luckily, Hank isnt that blood thirsty, and Stevell take a few shots before going down.
Fighting Skill: Ah, finally an area in which Cap wins. Caps trained for skill, he needs skill, he puts all his effort into working on fighting ability. Hes far more knowledgeable than Hank, and far more experienced than Hank. McCoy has ten years experience, and is an excellent fighter whos style is unique, and takes full advantage of his abilities. He doesnt, though, have all Caps tricks, or as much experience in one on one battles. Cap helped to train Beast, but I dont think that factors into the battle. McCoy has also been trained by Wolverine, in the Danger Room, and on his own. His style is nothing like it was five years ago, and many tactics which wouldve worked on him at his former strength level would be near useless now. Caps knowledge of Hanks style, if he relies on it too much, will be more harm than help. Cap is a far better fighter than Hank, but that is where his advantages end. On that note, fighting skill doesnt guarantee victory. Wolverine lost to the Hulk. Daredevil lost to Omega Red. The Punisher lost to Spider-Man. There is a point where power usurps skill, and Beast more than surpasses it.
Fighting Spirit: Captain Americas will is indomitable. He is on a level touched on only by Wolverine, Hal Jordan, Batman, and the like. Beast, however, is the next best thing. He fought off Mesmero when no one else could. He fought off Karmas mind control. As any Spider-Man fan will tell you, intelligence and humor dont mean you dont have the grit to win any battle you need to. Yes, Cap has an advantage, yes he is more ready to fight dirty, and yes it will help him. But in the big picture, this is a very minor concern. Will power wont stop a punch, will power wont make your punches connect, and will power wont keep you conscious when youve taken more punishment than you can handle.
On the shield: Throwing his shield is the worst thing Cap can do. Beast is faster and more agile than him, snatching it out of the air is simply not a problem. As AA pointed out to me the other day, Red Skulls done it. Taskmasters done it. Beast has done it. Also, while the shield is indestructible, the leather straps Cap holds onto are not. If Cap blocks with the shield, Hank can just grab it. He can rip it off his arm. He can wrench it from his grasp. If Hank wants the shield, its his, and with the advantage it gives Cap, hell want it. Its great for Cap to block with, as itll cover most of his head and chest from attack. Caps got it good there. However, He cannot block all of Hanks blows, given that hes dexterous enough to make multiple simultaneous attacks (All as efficiently as Cap can make one) and attack two or three times in the time it takes Cap to attack once. The shields good, but its not good enough, and it wont last long.
Scenario: This is how I estimate a Cap\Beast battle would go down. They square off for a second. Beast attacks. He leaps forward at Cap, rolling onto one hand, while he kicks at Caps chest. His other hand sweeps at Steves legs, and his remaining foot waits to block an attack. Cap ducks down, and rams the shield at Hanks supporting arm. Its stopped by his other hand, and Hanks kick turns into a flip. He lands on Caps back at 30 mph, and leaps away, still holding onto the shield. The leather straps snap, Cap is struggling to his feet, and Hank has the shield. Beast leaps to the wall, and bounces off. He speeds at 120 mph towards Steve. Cap, still shaken, sidesteps, and grabs Beast by the fur of his arm, and swings him out of control. Hank, though, quick and dexterous, snags Steves arm in return, and Cap goes flying along with him. Beast quickly pivots so that Caps on the receiving end of their crash into the nearest wall. Hank bounces off, and gives Cap another quick left to the jaw. Cap sprawls, but recovers, and ducks Hanks next punch. He gives Beast an uppercut, but Hank leaps above it, grabs Caps wings, and flings him against the wall. Cap sinks to the ground, and Hank stands back. Cap slowly rises. Hes bloody and battered, but refuses to give up. Beast holds the shield out grimly, and presses it to Caps hand, as he delivers a left that drives Cap a half inch into the wall. KO.
In reality, thisd probably be a little less one sided. Cap might get a shot or two in, Hank might not hit him that hard at the end, but this is more or less it. Cap can do everything right, and not win unless Beast screws up very badly somehow. It comes down to this: Each of them has a potential, and then they have how much they fulfill their potential. Caps potential is the absolute maximum human potential. Beasts potential is probably about twenty times that. Even if Cap is 100% efficient, Beast would have to be just 5% efficient for Cap to have an even shot. Beast is NOT 95% percent incompetent, not even in comparison to Cap. Beast wins.