Remember the rumor that Marvel was going to get rid of all its mutants? Last year, there was a big fear among some fans that, due to backstage politics between Marvel and Fox, the X-Men would be "No More'd" out of existence with a line-wide reboot that would effectively replace mutants with Inhumans.
That sounds terrible, obviously, but it's the kind of thing that nags at us when we think about film rights (as we so often do). However, while it can be fun to ponder a character's place in a fictional universe, there's little use in trying to discern the plans and motives (rumored or otherwise) of corporate executives. That way lies madness, Dear Reader.
Keeping the comics over here, the movies over there, and the business dealings far, far away is probably the best way to enjoy the stories. Yet, there's no denying that mutants and Inhumans can be viewed as similar entities in the Marvel Universe -- and, believe it or not, there are some decent reasons to at least readjust the X-Men's place within in it.
It's difficult to fight for a world that fears and hates you when simply be leaving the X-Men to hang with the Avengers means you're treated differently -- that is, significantly better -- by the public. Mutants are only hated until they're not, and I can see wanting to put the pressure back on the prejudice without resorting to another massacre. However, replacing the idea of mutants with Inhumans (or NuHumans, for those humans powered in the Terrigensis Bomb) isn't a one-for-one swap.
That's not to say they don't have similarities: Mutants are either a natural course of human evolution (because, naturally, humans would evolve to be able to fire energy through their eyeballs) or they're a more reasonable explanation, borrowed from Earth X.
According to that 1999 miniseries, by Jim Krueger and John Paul Leon (based on notes by Alex Ross), Celestials experimented with human life, and mutation is just part of a sequence of evolution. Mutants gain powers at random, and the next step is powers gained by belief from other beings. Because belief is the next step, it makes sense that a lot of mutant abilities have something to do with the personalities behind them. Mutants with major powers have issues with control. Vibrant personalities are explosive, and mutants manifest these powers (for the most part) when they're in their teens, when identity becomes distinctly important. Mutants are socially oriented to the point that their clash against humanity is a huge part of their narrative.
Inhumans are also an experiment with humanity, only by the Kree Empire. Intended as a way to jump-start Kree genetics and create a race of soldiers to defeat their enemies, the Skrulls, the Inhumans were ultimately abandoned when it was prophesied they would defeat their masters. Fascinatingly, the Krees didn't simply nuke them all from orbit, so a fully functioning culture and hierarchy was left to its own devices. Their main narrative is almost entirely interpersonal, as the line between the aliens and humans is clearly drawn. The Inhumans don't want to fit into human culture, and humans have no idea how to interact with the Inhuman nation. The fact that they are a nation, with their own laws and sovereignty, is the best part, as they play their roles to Shakespearean heights.
Mutants are a social mirror, with their parallels to real-world outsider cultures made abundantly clear. From powers that relate to our own insecurities, strengths and fears to the social standing between "us" and "them" and how each side can learn from the other ... well, the scholarly essays just about write themselves.
Fans have taken the X-Men in as our banner for the minority (racial or otherwise) and the disenfranchised. However, the Inhumans are distant enough from our own lives that they become a petri dish that's fascinating to observe. They are, in fact, inhuman; it's in their name. They don't operate within the norms of human culture, as they have their own rules and morality. Before they fell into New York City, they lived above us, on the Moon or in the Himalayas. As NuHumans are created, they either fall into Inhuman society or live apart from it. We read about the Inhumans for their Game of Thrones-like drama, not just in royal machinations and internal politics, but because our our own lives are so entirely distant from theirs, it's fun to watch it play out.
That's the fundamental difference between the Inhumans and mutants: We can't separate ourselves from the human/mutant connection any more than we could be Inhuman royalty. One offers us a way to work through our own differences, another is a way to observe, with some detachment, some of those differences play out. We can feel like King Lear in the twilight of our years, but we can't actually be King Lear. I can't see the Inhumans becoming more prominent than the X-Men in the Marvel Universe, as they both serve different storytelling purposes. Setting all rumor aside, the X-Men and Inhumans might be similar in scope, but entirely different through a reader's lens.