When Tony takes the nuke into the wormhole in Avengers, he has the change of heart after seeing what was on the other side. Before then he was an individual just learning about being a team player and he saw how that teamwork just barely allowed the Avengers and S.H.I.E.L.D. to come together to defeat it.

He knows it is going to take a lot more to defend against what he saw on the other side of the wormhole and wants to include gov'ts in on that, thinking it will pretty much take everyone to pull together to pull it off.

Bucky and HYDRA threaten that. They see a united world as one with unchecked power for those who lead it. They aren't the "good guys", but their position isn't a "bad" one either. Humanity is at its best and worst when it is united.

Tony essentially takes up the mantle of Cap's championing of the gov't and pulling together to defeat a greater enemy, but doesn't go all in, recognizing still that the gov't will abuse any and every means of empowerment granted to it.

Cap sees Tony as insanely intelligent, but incredibly naive, due to what he's seen Bucky have to go through.

Cap hasn't seen what Tony has on the other side of the wormhole. They have to have different points of view because of what they've been through and because of that, have to face off against each other.

And thus the Civil War, after which and even bigger and much more difficult war will have to be fought by those who've managed to survive the first.

And if Iron Man or Cap falls, the odds of surviving the greater war, let alone winning it, become incredibly long, if not impossible.