Comic Stark for much of his history didn't care about anything interesting to the general audience. He cared about his company, and he cared about the lives of soldiers in the abstract.
Very few people can relate to caring about a company. Most people will never see a million dollars in their lives, never mind a billion, so it's hard to make them care about saving a multi-billion dollar company. Oh no, if Stark fails, he'll...still have more money than I'll ever have.
While people do care about the lives of individual soldiers, most can't care about the importance of top-notch gear to those soldiers survival. It's too abstract. Also, it blends with their views on the military industrial complex, which taints what little they care about the subject.
Now I'll examine why Batman and movie Stark both work, despite their similarities to the failed comic Stark.
Batman cares about people with names and faces. He's out to stop murderers and rapists in alleyways. He also cares deeply about his sons, both adoptive and biological, and about his surrogate father Alfred. While he has a company, he's rarely shown to care much about it. It's mostly a plot-device to explain why he has so much time and money to devote to saving individuals with names and faces in dark alleyways.
That's much more relatable to the general audience than caring about billions of dollars or how your equipment will result in statistically ten percent less infantry casualties than your competitors equipment.
(And yes, you can make a case for Stark's big picture thinking saving more lives statistically than Wayne's, but that's completely irrelevant to the subject at hand, which is Stark's popularity with the general audience. This is about emotion, not statistical analysis.)
Batman's grief over his parents is a living thing, a very relatable source of angst.
The assassination of Stark's parents by Roxxon has been completely forgotten by the comics. I don't even remember when it was established anymore.
The result has been to make comic Stark incredibly dry compared to Batman.
Now on to movie Stark.
Movie Stark is filled with survivor's guilt and PSTD. The deaths of the people he's failed to save haunts him deeply. He mourns his parents, and their assassination is a major plot point.
The assassination of comic Stark's parents was established so long ago that I've forgotten the details, and never touched on again. This makes comic Stark a much blander, dryer character than movie Stark.
Movie Stark truly loves Pepper and Rhodey. Him being obsessed with keeping her alive is a major plot-point. This makes his heroism more awesome, and even makes his mistakes much more relatable and forgivable.
Comic Stark was with a different woman every issue. Stark didn't care about them, so why should the audience?
Movie Stark is a blatant super-genius inventor, and the movies aren't afraid to be campy about it. He has goofy robot pets. He does experiments that are awesome half the time, and blow up in his face the other half of the time. There's a real vibe that he's part of the tradition of movie super-genius inventors who cook their bacon and eggs with a Rube Goldberg machine.
This gives him routine scientific Crowning Moments of Awesome. It even gives him a way to screw up that is kind of awesome. His Frankenstein Monster Ultron, for example. Also his glancing at Maya Hanson's life's work, casually inventing the breakthrough equation she'd been searching for her entire career, scribbling it on a nametag, and forgetting about it. It's a mistake that causes the rest of the movie, yet it's still awesome!
One of the other posters talked about how comic Stark was less campy, but that ultimately turned into a weakness because the writers went too far with it, ultimately just making comic Stark feel bland and second-rate. While Thor was performing blatantly awesome feats of power and Captain America was performing blatantly awesome feats of inspiration and tactics, comic Iron Man was mostly this blandly "realistic" background figure.
Also, moving away from Stark being an inventive super-genius to him being mainly a businessman made the sort of mistakes his could make less relatable and forgivable. Super-genius Stark inventing Extremis' breakthrough equation or creating a Frankestein monster is divorced enough from reality and awesome enough that people can forgive Stark for it. Businessman Stark, though, has to make dreary, realistic, political mistakes that are much too close to home for most people to be forgivable. Not only don't people forgive those mistakes, many comic readers outright hate him for it.
Comic Stark's blandness even hurt his villains. Comic Stark didn’t care about anything interesting, so villains had nothing interesting to take from him. Batman cares about saving a random woman in a dark alley from being raped, and random people being murdered in dark alleys. Even if they are random, they are individual people with faces. Batman cares about something interesting, so villains have something interesting to take from him. Joker can hurt Batman, truly hurt him, just by killing one person with a name and face in a dark alley. That ability to truly, routinely hurt Batman makes Joker interesting.
Meanwhile what could The Mandarin or whoever take from comic Stark? He could have schemed to steal his company, which almost no reader cares about. He could have schemed to replace Iron Man’s armaments with cheap knock-offs that will statistically increase the danger of soldiers dying, but that’s too abstract to be interesting.