Most of my list has already been mentioned, but I am going to mention it again anyway:
First and foremost, I wanted a restructure of the "Elite" Four. I know it's a remake and I am just bitter that DPP programmers were lazy and didn't change it, but I really was hoping that the WEAK Four would have been scrapped and a TRUE tournament would have been added. You get all the badges and enter the Johto League Pokemon Tournament which is a field of 64 trainers. It goes 64 > 32 > 16 > 8 > 4 > CHAMPION. That is six rounds of battles with the option of healing your Pokemon inbetween rounds. You are allowed that because, unlike the WEAK Four, you won't know who your opponent will be until the next round and before each round you can have that little kid you find at the front of each gym and he can give you hints on your next opponents weaknesses. Total awesomeness.
While on the subject of "Elite" Four, I really wanted to at least face the Kanto "Elite" Four this time, and then Ash as the champ. Seriously, how easy would that have been to add? I just don't get GF sometimes. It's not like they are bumping up against the storage limitation of the DS or anything. That was completely odd as to why it wasn't that way in the beginning, but even more odd is not changing it when they had a terrific opportunity.
Just want to also add I wish they would acknowledge you as the CHAMP and that you were there to defend your title instead of it appearing as though it was your first time again. Again, one line of code is all that needed to be added for each battle. Simple.
I was VERY disappointed that the VS Seeker was not included, or at least a trainer (i.e. Gabby the Reporter of Emerald) that you could battle repeatedly. I guess I am left to constantly sending my Pokemon out on strolls to level up and then battle wild Pokemon for EV training. Much longer process.
Also disappointed that Secret Bases were not added either. There are just so many things about Emerald that make it still the best Pokemon game to date. I had 12 Secret Bases set up, 2 of each full of low level Pokemon that give 3 EV points of each stat. Between that, Macho Brace, and Pokerus, I could fully EV train 6 Pokemon in a matter of minutes. That is irreplaceable.
Now this one is a stretch, but I was hoping to see HGSS include the ability to track EV points. It could have been added on the stylus screen on the Summary page that has the stats on the main screen, and then move the moves to the blank screen on the third page. The game already tracks the EVs, so it wouldn't be that difficult just to show the stats on the Summary page.
That is all I will add now, but might include more later.