My favorite improvements in Emerald are all the overworld abilities that make breeding and finding pokemon a bit easier. It makes searching for a specific pokemon and breading the easier in gen 3. I have not found a list specific for emerald but you can read each ability's "outside of battle" effect to see which ones work in emerald.
https://bulbapedia.bulbagarden.net/wiki/Category:Abilities_with_field_effects 