I only have 4 badges so far, but I'll list mine here tomorrow, and update it as I progress through the game.
DexNav - This is by far my favorite thing so far, you can encounter pokemon with Hidden abilities, Egg Moves and see their potential much of time. Additionally you can spot pokemon with abnormally high levels for the area. Makes continuing to search for pokemon in a given area much more enjoyable than in games past as it feels like there is much more variety to it. It also has helped me being on a nuzlocke as I'm not completely blind as to which pokemon I get. It may be kind of cheating to do this in a Nuzlocke, but it's new so it can't be all that bad in that regard right?
The Match Call System - Making a return from the original R/S/E, this feature is the second best rematch system to only the Vs Seeker in my opinion. But it's FAR more realistic. I can't remember if rematched trainers in other games continue to scale based on how many badges you have, but I also love that. The format of the Match Call is much more easy to read and navigate through in this iteration of the game as well. You can also see what pokemon the opponent had the last time you battled them so you can tell if they're trainers you'd even want to rematch in the fist place. This, as a whole, is great for in game grinding and not forcing you to battle wild pokemon for hours on end.
Revamped Movesets - Some pokemon got access to moves they should have had for a LONG time coming. Others simply learned better moves at more reasonable levels. Either way it's a change I welcome with open arms. It also means less training for pokemon who want to breed for competitive battling. I'm shocked it took gamefreak so long to give Gyarados Crunch for example, especially with it's Mega Evolution being part Dark type. That's fixed now along with many others.
Secret Bases - I have not yet made my own secret base, but I have seen what can be done with them in this game and it is really exciting. Essentially being able to be your own Gym Leader? Sign me up right now! Now I can recognize my dreams of being a Ground Type Gym Leader and pound some scrubs into the dirt.
Storyline Fast Travel - This is so convenient, May or Brendan often times ask you if you would like to travel to X and it saves so much time especially considering all the HM's that have to be used in this game to navigate through the world. 'inb4 TooMuchWater/10'