The big difference I see is that most mutants have no idea they are mutants for the first dozen or so years of their life. Exceptions exist that are obviously mutants from birth (Nightcrawler, Madrox, etc.) but nine out of ten mutants have normal childhoods, until, bam, unasked for mutation on top of unasked for pimples and hair in new places. And then there's massive societal rejection, and often family rejection as well.
For Inhumans, prior to this NuHumans business, they were born into a city full of Inhumans, all different and with their differences being celebrated. They go into puberty eagerly awaiting their chance at the genetic lottery of Terrigenesis (and / or dreading getting a less wonderful transformation...), and when they step/float/slither out of the Terrigenesis chamber, it's to join a community of adults who've all been through the same thing, welcoming them to this society.
Before the NuHumans, the situation with suddenly becoming a mutant, and possibly losing all of your friends, family, getting kicked out of your school or church, etc., etc., and stepping out of the mist to become an accepted adult in a community of other Inhumans, is night and day.
NuHumans are indeed quite the mutant metaphor, and I'm not entirely thrilled with the idea, even if it allows for a much more diverse Inhuman community, since the Inhumans were originally all-white-Europeans. Thanks to both new Inhuman cities, and great numbers of people with Inhuman genes scattered throughout humanity, newer Inhuman/NuHuman characters can be black, Asian, Latino, etc. and have different cultural backstories than 'grew up in a hidden magic city full of super-people, waiting to become a super-people, like my parents and their parents, etc.' IMO, the utter same-ness of every single Inhuman from Attilan was probably the biggest problem with them ever really succeeding as a wider property.
And so, the entire point of the Inhumans has been sacrificed on this altar, making them more like mutants with the NuHumans, which, IMO, does neither mutantkind nor the Inhumans any favors.
It's like saying that chocolate is the most successful flavor, so every other flavor must now have chocolate mixed into it (or Wolverine is the most popular X-character, so every X-team has to have a Wolverine on it... Oh wait, that's a bad example...).