In movie/tv terms, maybe the manic pixie dream girl?
They’re seen differently because society has different expectations of men and women, but both have youthful qualities that put them outside the mainstream. In romcoms, it seems like the man child is either portrayed as someone who needs to be fixed by a woman who has her life in order (as in Knocked Up) or a wacky sidekick who gets in the way of a couple (You, Me & Dupree; Ted). A MPDG usually redeems a mopey male protagonist with her fun, spontaneous attitude, but doesn’t have many goals or desires of her own.
In tv and movies, the man-child can exist as an entity of his own. He has goals and desires and a love interest is secondary, even if society doesn’t really approve of him. A MPDG is more accepted by society- who doesn’t like a cute, fun, quirky girl?- but she’s all about the love interest.
The reason there are so many man-child guys in pop culture is because of their big, fun personalities. People still judge men on things like money and power, but confidence is at the root of both of those. It’s easy for a guy to be confident if he’s born into money and power. If he he has the same confidence even though he’s unemployed and living on someone’s couch, that dude must really have a set of brass balls.
In real life, you have to cooperate with other people and make responsible, unfun decisions. We don’t like man children for their immaturity, but because they just don’t give a fuck. For men who have responsibilities or aren’t very confident, it’s sort of an escapist fantasy. (Because confidence= power, so you can never have enough.) Most movie & tv writers, directors, execs, etc. are men, so we see a lot more escapist fantasies from the male perspective.
For women, for the longest time, finding the right guy was the only route to power and security. So when we see escapist fantisies from a woman’s perspective, it’s still usually centered around some guy. Like, uh, Twilight. That sells the fantasy that you can be bland and boring but some tall guy with a chiseled face will sweep you off your feet.
So if you’ve stuck with that long, convoluted explanation, Bella Swan is the lady version of the man-child.