The Sokovia Accords had merely been the fuse that ignited the civil war between Captain America and Tony.
In the end, the real reason had still become a dispute born of personal grudges.
After the Hydra incident, Captain America began to distrust the government and awakened a sense of self. Combined with Carter's death, which severed his last bond with the modern government, he finally began carrying out the justice he believed in.
Awakening a sense of self had both advantages and disadvantages. The advantage was that he finally knew how to think for himself and would no longer be foolishly used by others. The disadvantage, naturally, was that... he began to be influenced by his own emotions and became... less steadfast in his position.
A rational person inevitably possessed self-awareness from the start. Tony, for instance. Smart people always liked to think more, but his way of thinking was completely different from Captain America's. Their contrast became especially stark here.
The Sokovia Accords gave Moze, whose family had once been killed by the collateral damage from Avengers battles, an opportunity. He used the Winter Soldier—Captain America's friend Bucky—to trigger the civil war. But from that point onward, the nature of the matter began to change.
Captain America, who did not trust the government, firmly believed in his own thoughts and the truth. He believed Bucky must have had his own difficulties or some other reason, so he began resisting the government. In the movie's portrayal, Captain America's opposition always seemed to be in the right, imbued with strong heroic idealism.
But in the end, Tony gave up revenge, while most of Captain America's side were eventually willing to sign the Accords as well. Then the Avengers... split apart.
Leaving aside the Avengers' split and Captain America and the others