The idea of Rifttale is that Mt Ebott is located on a fault line, and all the hollowing out the monsters had to do made the mountain extremely unstable, and that problem only increased the more the underground's population grew. It's pretty dangerous underground in Rifttale what with the constant threat of cave in due to poor structural integrity and the fact that they don't really have much to work with to make their homes and businesses quake resistant. A fair portion of houses get rubble rained down on them and something always needs fixing.

All four sections of the underground remain, but the layout and placement are different. (And probably larger scale) For example, Snowdin is located near the top of the mountain since snow from the peak falls in through the cracks left in the roof of the cavern, and Hotland is at the very bottom of the mountain, where lava has surged up from the seismic activity. Both Waterfall and New Home are around the mid section, but are separated by a thick wall of rubble that followed a quake that happened maybe days before Frisk fell.

Sans

The man of many jobs adds two more responsibilities to the list rather begrudgingly in this AU. As one of the only Monsters capable of gravity manipulation and teleportation in Snowdin, he and his brother are tasked with being emergency responders in case of an earthquake as well as assisting with any initial construction steps in order to make bridges across newly formed gaps. Sans often gets called to other regions for his services too, but he really doesn't appreciate it.

Frankly he's convinced the mountain's going to crush them before they get the final soul they need, which is part of why he agrees to keep Frisk safe so easily. He doubts anything will change about their predicament with or without them. A No-Mercy route in Rifttale would see him saying things like "Couldn't have waited for the mountain to get us?" Or "I think I would have preferred a cave-in."

Something Rift!Sans doesn't really tell people is that he's blind in his right eye. When he and Papyrus were little, he shielded Papyrus from falling rocks during a quake, and one of them got him good just above his right eye. Most of the injury healed but it left a nasty crack in his socket which prevented him from actually being able to see. He can light the socket, and does so strangers don't needlessly worry about him, but it can't light up all the way like his left does.

Papyrus

Ah yes the sunshine boy. Personality wise he's not that different in this AU. He still wants to be in the guard and he's still confident and friendly, but sometimes he gets a little subdued and worried about Sans since Rift!Sans is much more obvious about his emotional/mental struggles. Another notable part is that he's being kept out of the guard mostly because he's more useful in the emergency unit. He's strong and capable enough to be a guard, but he's too soft, so they gave him a job that was helpful while playing to his strengths.... though like a silly he doesn't see the admirers he has and still hopes to become a guardsman one day.

Frisk & Chara

These two are interesting in this AU. Chara was around 13-14 when they died and spent years underground with the Dreemurrs before their death. They have a strong distaste for their own kind for several reasons, among which being the way they were treated (that led them to run away in the first place) and the fact that they forced such nice creatures into what is essentially a death trap.

They're a bit of a gremlin, but they're certainly not all bad like they consider themself to be.

Frisk on the other hand is a quiet child of roughly nine years. Adventurous and curious, but also blind. They hadn't seen the hole when they were exploring and ended up falling down. They rely on Chara to lead them around the more treacherous parts of the underground since they can't see it. (Chara is able to touch Frisk and tends to tug them by the wrist when guiding them.) Chara starts off really unhappy about guiding them, and depending on the route will even refuse to do so, not that it matters in a No-Mercy route. That's because in the No-Mercy route the Player is more of an active component. Frisk physically can't fight back on their own, and Chara would never help them do so. Essentially the Player is just an entity used by us creators to manipulate Frisk into a No-Mercy route since it wouldn't happen otherwise.

Asriel & Flowey

Personality wise the two really aren't all that different. Asriel is the slightest bit more skittish thanks to his timid nature and the uncertainty that comes with the constant tremors, and Flowey is the slightest bit more bearable. He still doesn't have much in the way of emotions, but it's pretty obvious their situation is bleak without him making it worse (As such, while he has tried No-Mercy in the past, most of his runs were seeing what effects smaller changes had. The volatile environment made it interesting regardless).

Though he's notably far more annoyed about losing his save and reload powers in Rifttale because he'd been in the middle of trying something when Frisk fell.

On the more technical side, Flowey doesn't tend to get around much when there aren't bridges over the chasms. While he can cross them himself, its a pain if they're too awfully wide. This means that the severe quakes tended to impede him a lot.

Toriel & Asgore

As a result of the meddling Flowey was doing, Toriel reclaimed her place as Queen and has been talking things over with her husband about everything that happened. Stuck in the ruins she hadn't gotten the news that Asgore amended his statement later on to say that they would only take the souls of humans that fell if they were hostile or after they died of natural causes. None of the children who fell were killed by monsters. (Many died in partial cave-ins or from falling into lava or the like, though so a quick death from a monster likely would have been preferable.)

Toriel still thinks that going out and just collecting the remaining souls they need would be faster, but Asgore argued a few points on that such as doing so likely angering the humans and giving them a further negative bias as well as the fact that whoever went to get the souls may well die like their children did.

They're still working things out but it's better than how it was before.

Toriel still spends a lot of her time in the ruins, but now its more out of a sense of duty since most of the children fell into the ruins first. She actually guides Frisk up until they meet Sans where she passes escorting them to him since he is more capable of ensuring their safety, and she needs to hurry ahead to make preparations for their stay and inform Asgore.

Alphys

Predictably not much different personality wise. She's actually one of the safer monsters since she never leaves her lab and it's reinforced to withstand rockfalls to some degree, not to mention the basement level with the true lab in it (though its hot as heck down there cause of the magma nearby.) Her primary goal is more geared toward keeping the underground as stable as possible so that they can hopefully last until they get the last soul. That's not to say the amalgamates don't exist, they do, but she's at least been looking busy as sort of an excuse why she hasn't answered back any of the letters. No time for mail when she's trying to save people, after all.

Undyne

Captain of the Royal guard, still. The guard does still exist, it's just less numerous and has more focus on recruiting strong members. After all they exist in case a hostile human arrives instead. So naturally Undyne is even more of a badass than normal, and she's ready in the event of a No-Mercy run. Even in a pacifist playthrough she's a bit wary of Frisk, notably not wanting to leave Alphys alone with them in case they were hiding their true intentions. This means she's the one who ends up escorting the human through the first little bit of Hotland after Waterfall, whereupon Mettaton would take over.

Mettaton

Considering a Human-killing robot was no longer needed, he was created with maneuverability in mind. The idea was that he could help others when needed, much like the emergency rescue portion of the guard, but Mettaton much preferred to focus on his career of course. He’s part of the reason Sans keeps being called to help out in other regions which gets him a healthy amount of resentment from the older skeleton.

Gaster

No major story involvement, but he does have plot importance. Gaster was the Skelebro’s father, and while he made them artificially, he did care about them a lot. Not much about him still remains and very few remember any details about him since he was quite a reclusive skeleton. He was close friends with the Riverperson however, so that’s who ended up essentially raising the two skeletons after Gaster fell into the Core. They provided for them despite being gone a lot.

Misc. Details

Children that stayed for an extended period were often called the “New Hope” a term coined when Chara first became a part of the Dreemurr family

Not all children opted to stay with the royal family, hence their belongings being stored in different regions

There were hostile humans that fell as well, but most met their fate in the lava and their souls were burnt up before they could be retrieved.

Frisk had fairly negligent parents. They weren’t outright abusive, but they really didn’t pay much attention to them.

Portions of the Waterfall part of the Underground have large nets bridging the gaps to catch the garbage that falls down that way nothing gets wasted falling into the chasms.

The Delta Rune on Sans’s coat is a reflective sticker. They told him he needed to wear something reflective for visibility’s sake and because he wasn’t willing to mess up his jacket, nor go without it, he cheated the system.

Papyrus wears a battle body just like normal, but he hot-glued some of the reflective material from human clothing that ended up falling down onto it so that he could both look cool and follow rules