Ultra Sun/Moon takes place exactly like in Platinum where most are basically still the same but with a few details redone and the story progressing towards Necrozma.
Necrozma has been circulating Earth to gather enough energy to go back to Ultra Space, since it was summoned to Earth by accident by AZ thousands of years ago. Since Solgaleo/Lunala are the only ones that can open portals, Necrozma infects either and controls them against their will and uses them to gather all Ultra Beasts to Earth for an invasion.
You'll have to battle Aether grunts/admins to collect Zygarde Cells and Cores.
Portal to Ultra Space in Resolution Cave.
Zygarde vs Guzzlord, Solgaleo/Lunala vs Necrozma, Ultra Beasts vs Island Guardians cutscenes.
There will be a smaller Battle Frontier similar to the BF of Platinum. New characters.
Battle Cave added alongside Battle Tree, gets more returning characters: Eusine, Karen, Zinnia, Steven, Phoebe, Crasher Wake, Cheren, Bianca, AZ, Diantha.
Island Guardians will be required to catch through main story.
You will have to defeat Aether Admins in five islands with Ultra Beasts: Melemele - Faba/Kartana, Akala - Helga/Pheromosa, Ulaula - Lest/Xruxitree, Poni - Wicke/Buzzwole, Aether Foundation - Aris/Celesteela.
Johto/Kanto Alolan Forms (works alongside Virtual GS): Wobbuffet (Psychic/Ghost), Smoochum/Jynx (Grass/Fairy), Remoraid/Octillery (Water/Fire), Ursaring (Normal/Psychic), Phanphy (Grass), Donphan (Grass/Steel), Marill/Azumarill (Water/Psychic), Shuckle (Bug/Grass), Ponyta/Rapidash (Water), Dunsparce (Dragon/Fairy).
Added later:
Gladion's role is exactly the same as it was in the original games. Part of Team Skull, a runaway from his mother. He gets a redesign though, and helps you tackle the Ultra Beasts in the later game. Most characters from Sun and Moon are completely similar to upcoming games, ala-Platinum like what I have said. Lillie, Plumeria, Guzma and Lusamine being the only exceptions. As for the champion, it's best to keep it hidden. One thing for sure is she's a returning character.
We don't return to Kalos. We do get to explore more of Ultra Space. An expanded version like the Distortion World.
Lance doesn't return in the games nor an appearance in the Battle Cave. And no, Johto starters doesn't get any special treatment. Some Johto Pokemon mentioned above are the only ones to get Alolan forms.
Crasher Wake is not redesigned. Cheren and Bianca's outfits are completely revamped to summer outfits, similar to Red and Blue.
Lest and Helga are twins, one with pink hair and the other one with blue hair.
S.S. Anne doesn't come back. The Battle Frontier's layout is exactly as the one from Platinum's Battle Frontier, only short by two Battle facilities. It's sort of a resort. National dex is present ingame from the start. Original Sun and Moon dex is expanded. We don't get starters, we do get Lugia and Ho-Oh and the beast trio post-game
Island trials are now post-game quests. Gyms are now present in the open lots in the original games and are needed to complete in the main story. Battle Cave is exactly like the Battle Tree but a layout of going underground and rental battles have returned.
Necrozma has been finding a way to go back to Ultra Space to contact its co-Ultra Beasts. Since Necrozma has become partially a Pokemon for living on Earth for thousands of years, it decided to invade Earth and make it a part of Ultra Space by opening portals using Solgaleo and Lunala.
Zygarde is a vital piece of the story. No reason whatsoever for it to be teased in the original games
There are definitely more places to visit but no new islands. The recurring islands have expanded its layout and more new areas to explore.
Ponyta/Rapidash are redesigned with mermaid tails, similar to a Hippocampus and is used as a Ride Pokemon in the upcoming games. It is also a signature Pokemon.
Petunia is a Bug-type specialist Gym Leader in the games.