Wi-Fi made a big impact I think, although I kind of miss the charm of trading and battling through a cable.
Breeding was another huge addition, and along with that, genders of Pokémon. You can't really imagine the games without them now.
Abilities were also something else that affected the game massively, something else to play around with and use as well as an effective moveset etc.
Natures, again, another mechanic that can help you if you find a Pokémon with the nature you're looking for.
As has already been mentioned, the physical/special split up of moves in Generation IV, it helped a lot of Pokémon gain more useful moves and make more use of moves they already had etc.
Some more minor things: FOLLOW ME FEATURE IN HGSS (seriously, that needs to come back! I know it started in Yellow kind of, but still); running shoes; the addition of different types of Pokéballs (like Dusk Balls, Net Balls etc) made it more interesting that just throwing around Ultra Balls after you've gotten so far in the game).
That's all I've got for now.