The rivalry between Captain America and Iron Man goes back a long way. While both have been Avengers since practically the beginning, they've found reasons to disagree several times since then. Sometimes it's only an argument, others it results in a shouting match, and, a few times, it's come to blows and all-out battle such as in the film Captain America: Civil War
But, why does this happen? They're both heroes and more-or-less have the same goals, so why do these two iconic Avengers find a reason to fight so often?
10 Man From The Past Vs. Man Of The Future
The most obvious difference between Steve Rogers and Tony Stark is that they come from different times and views on the world. Cap was born in the early 20th Century and has some nostalgia for the time, and Tony Stark was born in the post-war world and is constantly looking towards the future. Steve often refers to the modern era being far more complex than the simplicity of good against evil in World War II, and Tony wants to accelerate those "complexities" and make a new world even farther removed from the one Steve knew.
9 Idealism Vs. Pragmatism
While one might consider Tony's futurism innately noble, there is plenty of folly innate in that line of thinking. Tony's futurist philosophy endorses change and technological acceleration at all costs, even if Tony may not think of it that way. Tony thinks that technology will work everything out over time while Steve believes that positive change comes from the interior. Steve holds onto his values and altruism, and Tony thinks such things are fungible so long as the end result is positive. Steve's vision of heroism doesn't allow for such things.
8 Confidence Vs. Audacity
Captain America and Iron Man are both leaders, and their different ways of viewing the world will naturally shape their leadership. The two have different visions for the Avengers, and they've butted heads several times because of this. Steve has confidence in his beliefs, but Tony's confidence comes more from an arrogance and an audacity that he knows better and has the right to take the Avengers to that endpoint.
7 Class War
It's also worth mentioning that, eras aside, the two men came from very different backgrounds. Steve Rogers was the son of an immigrant who grew up in poverty in Brooklyn. Tony Stark grew up in the lap of luxury in Long Island. It's easy to imagine, somewhere in the recesses of his mind, that Steve Rogers resents this about Tony Stark. The Rogers had to scrounge and struggle just to survive while the Starks had more money than any human being could spend in a lifetime. It's hard to blame Steve for this.
6 Armor Wars
While disagreements and arguments permeated their careers as Avengers, the animosity began to manifest in the 1990's in two stories. One was the "Armor Wars" story arc in Iron Man comics. Iron Man discovered that his technology had been stolen and replicated by malefactors, and he began hunting down every armored hero and villain.
One of his targets was the Guardsmen, a corps of armored soldiers used by S.H.I.E.L.D. and the United States government to look over matters involving superpowers and the supernatural. Iron Man attacked the Guardsmen, believing that their armor may be stolen Stark tech. Steve tried to stop Tony, and the two came to blows.
5 Operation: Galactic Storm
The other story is "Operation: Galactic Storm" in the Avengers comics. The Kree and Shi'ar Empires are at war, and it's threatening to spill over to the Solar System In the midst of all this, the Supreme Intelligence of the Kree wreaked mass destruction upon his own people through the use of a Nega-Bomb, which he hoped would accelerate Kree evolution. Two Kree asked the Avengers to assassinate the Supreme Intelligence for the Kree. Steve said no, but Tony wanted to do it. In the end, the Avengers killed the Supreme Intelligence.
4 Civil War And The SHRA
This all finally came to a head with Mark Millar and Leinil Francis Yu's Civil War story, where, after a catastrophe caused by superhumans that left hundreds dead, the United States government passes the Superhuman Registration Act. This law requires all super-powered individuals to register with the government, submit their secret identities, and become super-powered cops on behalf of the government. Steve Rogers was against this, while Iron Man believed it to be the right move. Cap and a team of Secret Avengers went underground and continued to fight crime, and Iron Man was tasked with tracking Captain America down. This came to a series of open and destructive battles between Cap and Tony's teams. It resulted in the death of Goliath, aka Bill Foster, and, in the end, Captain America himself.
3 The Illuminati
Surrounding this and other major events in Marvel history is the Illuminati, which was made up of Iron Man, Doctor Strange, Mister Fantastic, Professor X, Black Bolt, and Namor. They manipulated several major events in Marvel history for the "greater good", including the Kree-Skrull War, the Secret Invasion, the Secret Wars, and Infinity Gauntlet. It was exposed when the Hood began hunting down the Infinity Gems, which the Illuminati had at the time. The secrecy infuriated Captain America, but he took an Infinity Gem for himself when the crisis with the Hood was resolved. However, he wanted the Illuminati to make their decisions in the open going forward.
2 The Illuminati Again
However, the Illuminati returned when the Incursions, two Earths from separate dimensions colliding and destroying their respective universes, began. This time, Black Panther organized the Illuminati and included Captain America in the planning. They reformed the Infinity Gauntlet to safely repel an Earth that was about to collide with theirs, but the Gauntlet and the Gems detonated in the aftermath. The rest of the Illuminati began discussing the possible need to destroy other Earths, but Cap wouldn't hear of it. So, Doctor Strange wiped his mind of any memory of the Incursions or of the Illuminati.
1 Original Sin, Axis, and Time Runs Out
However, Cap did remember thanks to the events of Original Sin, and he launched a global hunt for the Illuminati. At this point, Tony Stark has turned... "morally gray" thanks to the events of Axis, and he had been separated from the rest of the Illuminati. In the end, Steve was forced to work with the Illuminati for the greater good. The Avengers, the Fantastic Four, and the Illuminati built an "Ark" to save a select few from the Final Incursion, and Cap, wearing a suit of armor of his own, had one final bout with Iron Man to keep him from getting aboard the Ark while the world ended around them.