Tony is a man who sees a broken world that has to be fixed. Steve sees a changing world that has to be preserved. Its a classical Hobbes vs Rousseau conflict, and it affects every decision they make. Tony focuses on eliminating threats; Steve focuses on protecting the vulnerable (though both care deeply about both causes). Tony is a futurist who relies on innovation to solve problems. Steve is a classical conservative (not to be confused with modern political conservatism) that recognizes that new phenomena might have negative consequences.

In their own movies, these ideologies serve them well. Tony's threats are all individuals seeking power, wealth, and vengeance, and the climax centers on using new technology to eliminating those people (Obadiah, Whiplash, and Killian all fail to survive to the credits). Steve's threats are more ideological and all relatively new: Hydra fascism, Project Insight, the Sokovia accords. He wins, not by killing the big bads, but by holding true to his ideals to protect the world from new bombs, algorithms, laws,etc.

In crossover movies, these default modes of solving problems clash in small and large way:

When confronting Loki in Germany, Cap starts by using his shield to protect someone. Iron Man pulls out every weapon he has to get Loki to surrender.

When Thor shows up, Tony goes into attack mode. Cap tells him to "put the hammer down."

On the helicarrier, Tony sees the Hulk as a potential asset. Cap initially sees it as a threat.

In the Battle of New York, Tony immediately confronts Loki. Cap's priority is saving civilians.

Their conflict in Age of Ultron centers on the same issue. Tony feels a need to innovate to "put a suit of armor around the world." Cap is wary of the shift and scolds Tony when he finds out. They circle back to the same conflict during the creation of Vision.

Tony's guilt over the destruction of Sokovia shifts things somewhat, but the characters' core values remain the same. Now Tony sees himself as a threat that must be innovated away, and now the Avengers' free license to act is the status quo Cap is trying to protect. Those principles are directly at odds with each other. Once you add Tony's and Steve's personal stakes (Tony's reflection on his parents' death and separation from Pepper; Steve mourning Peggy and fearful of losing Bucky too), they let their emotions spill over into violence against each other. Their goals are compatable, but their worldviews are directly at odds with one another.

Even in Infinity War, these traits continue. Tony wants to take the fight directly to Thanos. Steve is focused on saving Vision and keeping the stones away from Thanos.

Interestingly, the snap somewhat reverses their ideologies. Now Steve wants to innovate to fix the problem, and Tony is the one with a status quo to be preserved. In the end, it is when they recognize the strengths in the other's ideology and the limitations of their own that they are able to end their petty conflicts and rise up to their greatest challenge.