Originally Posted by
vitruvian
As far as the in-universe explanation goes, it's pretty simple:
1) Superpowered mutants get their powers from the X-gene. It appears to be a single gene mutation that somehow gives different individuals born with it (either by mutation or inheritance from X-gene positive 'mutant' parents) wildly different powers. The potential for the X-gene to crop up in the overall human population is believed to be a legacy of genetic tampering long ago by the Celestials, alien space gods who also created the variant human races of the Eternals and Deviants. Reasons for the mutant population exploding in recent decades are unclear, with theories ranging from exposure to radiation from nuclear tests to pollutants as mutagens to it just being time for the next stage in evolution (never mind that the potential for the X-gene was artificially introduced by mile high aliens) to it being the Celestial plan for Earth (never mind that the Fourth Host gave Earth the thumbs up while still populated mostly by baseline humans rather than mutants).
2) Inhumans can get superpowers from exposure to the Terrigen mists (rather than dying from them) because they were genetically engineered and tampered with by a different alien race, the Kree, rather more recently - but still a long, long time ago. This genetic tampering may have built on the stuff the Celestials had already done to the human genome. For a long time it was thought that the Inhumans were all sequestered from the larger world in their Hidden Land, but it's recently turned out that there is a minority of people around the world with enough Inhuman ancestry from groups that wandered off and interbred with other peoples that they can successfully undergo Terrigenesis as well, and a whole bunch of them recently got powers from Black Bolt and Maximus exposing the world to the mists (but in a low enough dosage it didn't kill regular humans without the Inhuman genetics).
As for what makes an Inhuman different from mutants, prior to the Terrigenesis, the 'pure strain' Inhumans that have been living in the Hidden Land (and then the Himalayas, and then the Moon, and then in Kree space, and then floating above NYC, and now kind of scattered or in the crashed city) are supposedly stronger, healthier, and longer-lived than regular humans even without exposure to the mists, but they don't get powers on their own. Mutants develop their powers and other mutations spontaneously, usually but not always at puberty. As for all the recent Inhumans who thought they were just regular humans and turned out to have enough Inhuman ancestry to undergo Terrigenesis, I don't think we've been shown or told if they were any different from normal before the mists touched them.
As for what the franchise stands to lose/gain, I think it's mainly an additional franchise, not a replacement, as mentioned above. Within an in-universe context, I suppose it would make some sense to say that what the Kree managed to do was just create a modified X-gene that only got activated by Terrigen, making Inhumans just a different type of mutant and the two groups in some sense one and the same.