Graphics are significantly better, everything is colourful and the dodgy sprites of Gen 1 are no longer.
17-type mechanics, Gen 3 game mechanics of things like EVs, IVs, natures, abilities, shiny/PKRS, separation of Sp. Atk and Sp. Def, reclassification of some moves like Bite, and retyping of Pokémon like Magnemite. No Mew glitch, no Missingno, more bag space. Sadly no time mechanics, so forget about using an Espeon or Umbreon. NPCs have names, and you can play as either male or female character. EXP share is a held item now. Limited number of Gen 2/3 Pokémon available, including added evolutions and pre-volutions. One of the Legendary Beasts available after postgame, (save your Masterball for this as it will disappear permanently if it flees using Roar.)
Elite 4 and Champion can be rematched at higher level. Their teams also make use of newer Pokémon.
Some trainers can be battled again using the VS seeker.
2 Day Care facilities. The one in the Sevii Islands can take 2 Pokémon and allows for breeding.
Some moves function differently, for example binding moves no longer restrict use of attacks, and Venusaur can no longer exploit the Toxic & Leech Seed glitch.
Sevii Islands - accessed after badge 7, and also added postgame featuring Team Rocket. New functions including a move reminder found here. This is also a decent spot for grinding levels by constantly rechallenging trainers, and the new home of Moltres.
I'll add if I think of anything more.