The linguist who created the Grounders' language for the TV show (David J. Peterson) has addressed aspects of this question via a blog post on Tumblr How did the Grounders' language change so fast?
The difference between the Ark/Mount Weather and the Grounders is non-linguistic. Specifically, society remained pretty much intact on the Ark and in Mount Weather. Yes, they had to change the way society worked, but, for the most part, they were all safe and could retain social institutions like education, recreation, etc. In addition people had the ability to live quite a bit longer in both locales, provided they followed the rules. There’s certainly no one there who will have been alive during and remembered the old world (or most likely? Perhaps if they were very young, they could’ve survived in Mount Weather), but I bet there are a few that are one generation removed. I doubt if that’s true on Earth. Everyone on the ground had to worry about the very basics of survival. All social institutions were overturned. Mortal danger became a real part of everyone’s everyday lives, and illnesses could run rampant. Frankly, it’s quite surprising anyone survived at all. (And, as we’ve seen, not everyone survived perfectly; some have mutated.) It’s my guess that there are few if any second generation Grounders alive.
The result is that not as much information is passed from generation to generation. Furthermore, innovations from the younger generations are much more likely to stick around if there are fewer older speakers to gainsay them. There’s less push for them to assimilate to any cultural norms if (a) they’re not being passed on as readily, and (b) they’re just as likely to be creating it on their own. Thus the language evolves a bit more quickly.
Recall that with Latin, as much as society back then was less technologically advanced, the social institutions were just as strong as they are now. There was no societal collapse the way there was on Earth 97 years before the setting of The 100. It’s a different environment.
The last piece of the puzzle is a bit of fiction we concocted. Given that we’re in a small geographical area, there are features that are present in the modern language that are a direct result of conscious change on the part of early speakers. In the chaos that prevailed in the early days, there were direct innovations created so that survivors could determine if someone new they came across was one of them or wasn’t. Those that organized early developed vocabulary that would allow them to easily identify other group members—in addition to being able to communicate with group members without giving away what they were talking about. It was basically a code. Many of these old code words eventually became the new words for what they referred to—defeating the original purpose of the code, of course, but by then it no longer mattered. The fittest had survived.
Lastly, the warriors specifically retained their fluency in English in order to be able to understand everyone else (e.g. the Mountain Men). They’re reluctant to speak English in front of outsiders, though, because they don’t want to tip their hand. This is why it took Lincoln so long to actually speak in season 1: He was gathering information on the 100.
(Bold is my own emphasis.)
So in summary, the Grounders were subject to unique environmental pressures, their population had different sociocultural characteristics and priorities, and they made strategic choices that all contributed to the development of a distinctive language.
The Ark and Mountain Men populations were not subject to these same conditions.