Better parts of HG/SS: Safari Zone, graphics, Pokeathelon, easier to find legendary beasts, Goldenrod Department Store lottery, Blackthorn Move Relearner, and more Apricorn trees.
Better parts of GSC: music, Game Corner casino games, Eusine plotline in Crystal (I don't know what happened to him after you catch Suicune in the remakes), being able to rebattle the trainers on the SS Aqua, your mom changing the Pokegear clock for you for daylight saving time, and being able to delete NPC trainers from your Pokegear phonebook.
I know why the first two things I mentioned about being better in GSC were changed for the remakes, but I don't know why Game Freak changed the other stuff when it was best the way it had been before.