-Festival Plaza is essentially your new online communication hub for Wifi battles among other things (missions as well)
-Pokemon Amie is now called PokeRefresh, where you can pet them, feed them pokebeans, and care for them post battle to increase affection and enjoyment values.
-Battle Royal - a 4 player mode where each player sends in one of three Pokémon they bring to battle. In it, they can target down any of the other Pokémon on the field. This mode is a Free-For-All, so it's every player for themselves. Spread moves have their damage reduced by 50%. Mechanics for Double Battles such as attention diverting through Follow Me or the new move Spotlight are in effect here.The mode can be played in a special facility in single player, or over multiplayer using the Festival Plaza communication modes.
-Poke Pelago is an app that you use to harvest Pokebeans to feed your pokemon (and use for exchanges), expand small islets for pokemon in your PC to explore, and grow harvests of berries (which are useful for when you need to quickly reduce EV points by feeding pokemon berries... I'm still looking into that process myself), among other things. Hope someone can explain Poke Pelago more in depth than me!
-HM moves do not exist in Alola (you can obtain them as TMs, but I don't know the extent of it past Fly), instead you have a pager equipped to call forth Pokemon to help you with tasks HM moves used to do for you such as: fishing, Surf, pushing boulders, smashing rocks (in and out of water), dowsing for items on the ground, walking on rocky terrain, Fly to certain areas on the map (and can fly between islands), et cetera
-Gym Leaders have been replaced with Captains+Totem Pokemon (Island Trials) and Kahunas (Grand Trials). Elite 4 is historically new to the region, but completing all the trials and defeating all the Kahunas earn you the title Trial Champion. Defeating Captains and Kahunas reward you with Z-crystals for a particular pokemon type as the Alola verson of 'badges'.
-Your Pokedex is essentially a Rotom inhabiting it-- you can also take pictures of roaming pokemon at certain prompt areas and save pictures to your DS.
-Ash-Greninja/Battle bond abilities. I don't know much about this one.
-Z-Crystals must be held by Pokemon with moves of the same type as the crystal they hold (or have a Z-Crystal specifically for them) to activate their Z-moves; these essentially activate a super powered move that deals quite a bit of damage-- but if you miss, you cannot do it again because Z-moves are limited to 1 use per party per battle.
-No Megas for us (?); instead, the starters each obtain their own unique Z-Crystal for their final stage forms to use.
-Some pokemon appear in a different form (Alola formes); while you can still see the same species in the usual form, those caught in Alola will evolve to their Alola form. This usually implies changes in stats/type/movesets compared to their non-Alolan versions.
-Zygarde cores and cells are collected throughout the region to power up Zygarde into different forms, iirc. I may be wrong.
-If you keep the titular legendary in your party and complete the rest of the game afterwards, you apparently unlock a secret cutscene. XD
-SOS Calls - Wild pokemon at the end of a turn may or may not attempt to call forth an Ally pokemon, which may or may not appear. People have used these for EV training, Shiny hunting, and EXP grinding.
-At cafes in pokecenters, you are rewarded Rare Candies if you feed your pokemon Pokebeans often enough.
-The daycare is available early on in the game, so you can get to work on breeding if you wish.
