I didn't have my DS connected to the Internet at the time of Pokemon Black and White, so I'm unsure to how good it was (Either way, they're my favourite Pokemon games).
I did think the Poketch was interesting, despite me not being a large fan of Gen 4. It literally had a calculator, I think a friendship checker, an app to see if there's an egg at the Day-Care and a coin flipper! It was kinda cool.
The PSS was very accessible, being able to see every player you've interacted with, whether such players are online, as well as random players in general. Super Training and Amie were nice additions too (although from when I last played X, a while ago, I used neither). The Pokenav was pretty useful, in that you could track what you've caught from an area without consulting the Pokedex, whilst also serving as a shiny-hunting mechanism.
Gen 7's Rotom Dex I had never used, specifically the USUM version, with the free healing items, and capture rate increases. They're useful, but I never really needed them. The Sun and Moon bottom screen was ok, it showed the next place you were required to go to which is a first in the series, and had a map of the area.
I'd say my favourite would be the ORAS bottom-screen or Poketch. I say this because ORAS combines the XY bottom screen along with a way to track caught Pokemon, and find shiny Pokemon. I like the Poketch due to all the applications it has, and in general adding a layer of convenience to the game, albeit (in some cases) unconventional ways, akin to the aforementioned coin flipper. (I'm not sure when you'd need a coin flipper in a Pokemon game, but it's something extra.)