HGSS. FRLG tried to stick WAY too closely to the originals, even to the point of excluding Yellow additions (Jessie/James, Surfing Pikachu, following starter) as well as standard features that had existed since Generation 2 (day/night, berries, breeding, newer pre/evolutions). It's amazing that they even allowed you to play as a girl. If I wanted to play old school Gen 1, I'd just whip out my Yellow cartridge; at least, there aren't any annoying natures/abilities to worry about.
(Seriously, who else wants to see a new Gen 1 remake, one that's done RIGHT?)
HGSS, on the other hand, made an effort to mix things up by adding new areas (Routes 47-48, Safari Zone, Battle Frontier), restoring areas in Kanto (Viridian Forest, Cerulean Cave, Seafoam Islands), redesigning many cities and routes, merging the Gold/Silver and Crystal storylines, adding brand new features (walking Pokémon, Pokéathlon), fleshing out returning characters (Silver and the Rocket Admins), and so much more. They also had some gorgeous graphics, to boot.
As for ORAS, aside from one playthrough of Emerald, I'm not a big Hoenn person, so I won't say much about those. They appear to be much better than FRLG, though, so I'll give them that.