This is one debate that will go on forever.
Without weapons, fighting in a ring, I would say Cap has the edge given that he is stronger and has greater endurance.
With weapons and in an urban environment, Bats has the advantage given his wider variety of offensive tactics (utility belt) and greater stealth.
Captain America (Steve Rogers)
Height: 6'2" Weight: 240 lbs.
Rogers in the regular Marvel Universe has no superhuman powers, although as a result of the Super-Soldier serum, he is transformed from a frail young man into a "perfect" specimen of human development and conditioning. Captain America is as intelligent, strong, fast, agile, and durable as it is possible for a human being to be without being considered superhuman. The formula enhances all of his metabolic functions and prevents the build-up of fatigue poisons in his muscles, giving him endurance far in excess of an ordinary human being. This accounts for many of his extraordinary feats, including running a mile in a little more than a minute. Furthermore, his enhancements are the reason why he was able to survive being frozen in suspended animation for decades. Rogers is also unable to become intoxicated by alcohol and is immune to many diseases.
Mentally, Rogers' battle experience and training make him an expert tactician and an excellent field commander, with his teammates frequently deferring to his orders in battle. Rogers' reflexes and senses are also extraordinarily keen. He is a master of boxing, jiu jitsu, and judo, combined with his virtually superhuman gymnastic ability into his own unique fighting style with advanced pressure-point fighting. Years of practice with his indestructible shield make it practically an extension of his own body, and he is able to aim and throw it with almost unerring accuracy and even ricochet the shield to hit multiple targets. He is extremely skilled in hand-to-hand combat, sometimes taking on and defeating foes whose strength, size, or superpowers greatly exceed his. In the comics, he is regarded by other skilled fighters as one of the best hand-to-hand combatants in the Marvel Universe.
Rogers has vast U.S. military knowledge and is often shown to be familiar with ongoing, highly-classified Defense Department operations. Despite his high profile as one of the world's most popular and recognizable superheroes, Rogers also has a broad understanding of the espionage community, largely through his ongoing relationship with S.H.I.E.L.D.. He occasionally makes forays into mundane career fields, including commercial arts, education (high school history) and law enforcement.
Captain America is revered by most of the superheroes in the Marvel Universe, filling the "leadership" role which Superman takes on in DC Comics. He is considered a living legend, and many characters (particularly Spider-Man) idolize him.
Weapons and equipment
Further information: Captain America's shield
Captain America uses several shields throughout his history, the most recognizable of which is an indestructible discus-shaped shield made from a vibranium/steel alloy (not adamantium-vibranium as sometimes erroneously stated). This alloy is accidentally created and never duplicated, although efforts to reverse engineer it result in the creation of adamantium.
Captain America's uniform is made of a fire-retardant material, and he wears a lightweight "duralumin" chainmail beneath his uniform for added protection. Originally, Roger's mask was a separate piece of material, but an early engagement had it dislodged, thus almost exposing his identity. To prevent a recurrence of the situation, Rogers modified the mask with connecting material to his uniform. This also had the fringe benefit of extending his armor to cover his previously exposed neck. As a member of the Avengers, Rogers has an Avengers priority card, which serves as a communications device.
Batman (Bruce Wayne)
Height: 6'2" Weight: 210 lbs.
Batman is a superhero despite his not having super-powers. His resourcefulness, insight, and years of rigorous training make up for the absence of any other special abilities.
Physically, he is at the peak of human ability in dozens of areas, notably martial arts, acrobatics, strength, and escape artistry. Intellectually, he is just as peerless; Batman is one of the world's greatest scientists, criminologists, and tacticians, as well as a master of disguise. He is regarded as one of the DC Universe's greatest detectives. Rather than simply out-fighting his opponents, Batman often uses cunning and planning to outwit them.
Batman designs or modifies the majority of costumes, equipment, and vehicles he uses as Batman, producing them through various divisions of Wayne Enterprises, including Kordtronics. At various times, characters such as Oracle, Harold, and Toyman III create, modify, or repair Batman's equipment. Additionally, sometimes Batman adapts or reverse-engineers the technology of other villains and heroes, such as Mister Terrific's T-spheres.
Over the years, Batman accumulates a large arsenal of specialized gadgets (compare with the later James Bond), the designs of which usually share a common theme of dark coloration and a bat motif. A notable example is Batman's primary vehicle, the Batmobile, usually depicted as an imposing black car with large tailfins that suggest a bat's wings; another is his chief throwing weapon, the batarang, a bat-shaped boomerang/throwing star. Batman's other vehicles include the Batplane (aka the Batwing), Batboat, Batmarine, and Batcycle.
In proper practice, the "bat" prefix (as in batmobile or batarang) is rarely used by Batman himself when referring to his equipment, particularly after some portrayals (primarily the 1960s Batman live-action television show and the Super Friends animated series) stretched the practice to camp proportions. The 1960s television series Batman has an arsenal that includes such ridiculous, satirical "bat-" names as the bat-computer, bat-scanner, bat-radar, bat-cuffs, bat-pontoons, bat-drinking water dispenser, bat-camera with polarized bat-filter, shark repellent bat-spray, and bat-rope. In one episode, Batman and Robin stop by an outdoor hamburger stand which sells "bat-burgers", beef sandwiches supposedly named in his honor. The storyline "A Death in the Family" suggests that given Batman's grim nature, he is unlikely to have adopted the "bat" prefix on his own.
Batman keeps most of his field equipment in a signature piece of apparel, a utility belt. Over the years it is shown to contain a virtually limitless variety of crimefighting tools. Different versions of the belt have these items stored in either pouches or hard cylinders attached evenly around it.
In some of his early appearances, Batman uses side arms (see especially Detective Comics #32, September 1939), but he uses them less over time, later eschewing their use because a gun was used to murder his parents. Some stories relax this rule, allowing Batman to arm his vehicles for the purpose of disabling other vehicles or removing inanimate obstacles. In two stories, The Dark Knight Returns and The Cult, Batman used machine guns loaded with rubber bullets rather than live ammunition. In the 1989 Batman film, firearms figure more prominently in the Dark Knight's arsenal; machine guns and grenades are mounted on the Batmobile, and missiles and machine cannons on the Batwing.