There are no words to describe my feelings about this relationship. But I’m going to try.
First of all, their parallels. Both geniuses, top of their field. Both suffered an accident that physically changed them, forever, and not in a wholesome Spider-Man kind of way. Both try to do what they can to help others despite their own issues; Banner heals people, Tony works on developing clean energy. And both struggle, in their own way, with duality; Tony and Iron Man, Bruce and the Hulk. Two identities, one body. Only difference is Iron Man’s bad side is Tony.
I mentioned somewhere that Tony sees a bit of himself in Banner because they both have a monster inside them that they can’t control, a creature that springs fully formed from the id, the base impulses and the nasty stuff at the back of the mind. Bruce’s is a giant green rage monster. Tony’s trashed a party in Iron Man 2. Banner has a control over his that Tony hasn’t quite achieved yet; don’t think I didn’t notice Tony pouring himself a whiskey when confronting Loki. Tony is envious, fascinated, and most of all, impressed by Bruce’s control.
So he doesn’t walk on eggshells around Bruce like the others, because that’s not what Bruce needs. Tony sees Bruce’s restraint, sees the quiet, brilliant man making self-deprecating jokes in the corner of the room, sees the way people look at him like he’s going to snap any second, and thinks “nope”. Tony does what no-one else aboard that Helicarrier does. He trusts him. He makes jokes and jabs him and teases him and above all, treats him exactly how he would treat anyone else— he has a great regard for Bruce’s brilliance, and tells him so, but he doesn’t try to ignore the Hulk in the room. When he says “wow, you’ve really got a handle on this, haven’t you?” he’s not saying “gosh, it’s incredible you haven’t snapped yet and killed everyone on board” he’s saying “I know you have a handle on this, you wouldn’t be here if you didn’t, so I’m gonna poke you with this sharp object to prove it”. And you can see Bruce relax, and smile, and trust him back.
But then Tony goes even further, and invites Bruce to come to his R&D department. I’m pretty sure the two of them drive off together in Tony’s car at the end of the movie to do just that. And, okay, sure, Bruce is smart, but Tony’s tech is his baby. How many people get invitations to come and see his work? He invites Bruce because he recognises his brilliance, yes, but there’s another reason. He’s inviting Bruce to come down and work with him after this is over. He’s giving Bruce something to do next, a purpose, an alternative to disappearing into the ether to be alone with his monster. Tony knows from experience that being alone with your issues doesn’t end well, so for what’s only the third time in his life he extends the hand of friendship to a guy he’s known barely an hour.
And then, he tells Bruce to let the beast loose. Not just because they need him to fight, but because it will help him. If Bruce can take this thing that he sees as a curse and turn it into a gift, well, that’s going to lift him out of a very dark place. I’m not saying Tony knew about Bruce’s attempted suicide, but I think he had a suspicion that Bruce had been, in his words, “low”. So he encourages Bruce to take all that crap and pain and the Other Guy and use him to help people; after all, that’s what he did.
And it pays off. Nobody— nobody— thinks Bruce is going to turn up for that final battle. You can see the look on Natasha and Steve’s faces when Tony asks if Bruce turned up yet. They’ve counted Bruce out. Guy’s a mess, right? He’s too volatile. Doesn’t play well with others. He could never work as part of a team. No-one thinks he’ll come through when it matters. Except Tony. He has faith in him, and that faith is rewarded. It’s no wonder the Hulk is the one to catch Tony. Tony’s the one who helped let him out. He’s just returning the favor.