It's pretty easy for superhero fans to compare Marvel Comics' Captain America to DC Comics' Superman. They wouldn't inherently be wrong, as they are both proponents of the American Way seen as boy scouts by their heroic colleagues. In fact, when the Amalgam Universe was created, Cap and Superman were even combined into one hero. However, an alternative and perhaps better comparison to make would be the idea that Captain America is in fact closer to Batman than he is to the Man of Steel.
Captain America's origin began with Steve Rogers in a position of weakness, due to his fragile body. He felt helpless, unable to serve in a war that he desperately wanted to help end. As a result, when the opportunity came to be more and to transform himself into a being of power and strength, he took it, becoming the Super Solider known as Captain America. As Captain America, he went to work righting wrongs, seeking justice, and fighting for those who could not during WWII and into the present-day Marvel Universe.
The same can be said for Batman's origin. Young Bruce Wayne felt helpless to save his parents from the mugger that killed them. As a result, Bruce took every opportunity he could to transform himself into a being of power and strength, making sure he was at peak performance to become Batman, the Dark Knight. As Batman, he likewise sought justice and fought for those who could not in his own city of Gotham in the DC Universe.
The comparisons don't end with their origins however. Both Captain America and Batman are similar in that they've rarely deviated from their focus on justice, struggling but never failing to adhere to their own moral absolutes. Their consistency is what makes them both crusaders, refusing to stray from what they stand for, and what dictates their cautious but confident tactics. Furthermore, their lack of superpowers in comparison to their allies limits them both to the peak of human performance in their respective realities. Regardless, they both carry similar roles for their respective teams of the Avengers and Justice League - more often than not, they are their team's leader, tactician, and strategist, always counted on to come up with a plan to save the day. They've also proven that they're both evenly matched fighters, as confirmed in the JLA/Avengers crossover, and while it's easy to forget that Captain America's shield is just another tool, it's worth noting that - like Batman - his standard approach is direct physical confrontation aided by a simple thrown weapon.
Again, it would be easy to compare Captain America to Superman, and likewise Batman to Iron Man (they both have insane amounts of money, after all). The key difference, however, is in how they experience the danger implicit in their roles. While Iron Man and Superman can easily take a lot of direct punishment and have advantages like drones or superspeed to help save civilians, Batman and Captain America are ultimately far more aware of their vulnerability and mortality. Walking into any situation, Batman and Captain America are both, by necessity, checking every angle and considering every possible tactic, because a surprise attack doesn't mean a bullet glancing off their shoulder, it means death. Batman considers himself a soldier in a war against crime, and Captain America likewise brings the eye and priorities of an experienced soldier to superheroics. So while Steve Rogers has the same pleasant demeanor and down-home values as Superman, his stern tactician's demeanor in battle makes him much more similar to DC's Batman than their big, blue boyscout.