One thing that is now different that if you're playing on normal difficulty, you can be killed in battle and choose to resume starting on the floor you last reached with the HP/MP, EXP, items and money that you had when you first reached that floor. Anything gained on that floor prior to your death is gone though. Because of this system, savepoints at the end of dungeons just before the major shadow fight are removed, as is floor skipping.
The Persona fusing system is also tweaked so you can manually assign skills fused persona gain, use a search field to find a specific Persona you want, so that boss fights against certain elements are easier to deal with.
Oh, and treasure chests now respawn as soon as you leave a dungeon floor, meaning you can reacquire more healing items, MP restoring items and more. Also when you defeat shadows and gain cards, you can choose the card you want, no more roulette. There are new feature cards too like partial restoring of the whole party's HP/MP.
All of this means level grinding is no longer really needed and the dungeon crawling is a lot more enjoyable than in the original P4. When I beat Persona 4 on PS2, I clocked 178 hours into it. So much of that was due to grinding. Persona 4 Golden feels liberating that you can focus on all the good bits like the social links and such and cut the level grinding time by two thirds.
There are a TON of changes. For a full list, check here:
http://megamitensei.wikia.com/wiki/Persona_4_Golden