I think the crux of the difference of choice between the two comes down to the issue of accountability and responsibility.
Cap wants to be held accountable for his actions and decisions.
Steve: Tony, you chose to do that. If we sign this, we surrender our right to choose. What if this panel sends us somewhere we don't think we should go? What if there's somewhere we need to go and they don't let us? We may not be perfect but the safest hands are still our own.
They are his powers to use as he sees fit, but he recognises that anything he does with them is his responsibility, if he makes a bad call, he takes the fall for it.
Steve: We are if we're not taking responsibility for our actions. This document just shifts the blame.
He's also uncomfortable with being used at the whim of those in power, without any say in whether he agrees to that cause or not.
Steve: No, but it's run by people with agendas and agendas change.
He's been shaped by the events of The Winter Soldier, where he sees what happens when there are rotten apples at the top and that the rot isn't always visible until it's too late. Consider particularly the beginning of The Winter Soldier where he's sent to rescue hostages only to discover that's a secondary objective to getting information on Project: Insight, which ultimately turns out to be Hydra's grand scheme. Steve is also highly influenced by the events of World War 2. While Tony would think "you register a gun, so why not a person with destructive capabilities", Steve has seen what happens when you put people on a register because of who they are. For Steve the horrors of World War 2 and the holocaust are only a few years old, and that has to have a great impact on his views.
Tony on the other hand sees things from the other side. He's seen what his weapons have been used for (Iron Man 1), he's seen the results of his own (well intentioned) actions and the devastation they cause (Age of Ultron) and he doesn't want that weight on his shoulders.
Tony: That's good! That's why I'm here. When I realized what my weapons were capable of in the wrong hands, I shut it down. Stopped manufacturing.
He doesn't believe that kind of power should sit in the hands of individuals, because not everyone will use them for good, and quite critically, even those trying to do good can often find unintended consequences (like releasing Ultron on the world). He's also astute and has seen that if they don't agree to this, something much worse will be forced upon them later.
Tony: If we don't do this now, it's gonna be done to us later. That's the fact. That won't be pretty.
In short, Tony has been shaped by what happens when people use their powers without oversight, and Cap has been shaped by what happens when that oversight doesn't have good intentions.