What Reaper of Souls Does Differently
The Auction House (both Real Money and gold-powered) is gone. Blizzard has openly acknowledged that the AH, while an interesting idea, ultimately worked against its own goals for the game. Farming the AH isn't fun, and Blizzard recognizes it. High-end equipment and recipes can be shared among party members for up to two hours after you find an item, but you can't sell anything above a rare item to another player.
The Pandemonium Fortress
New Itemization: Along with Smart Loot, the new system focuses explicitly on a set of core "useful" statistics. The number of points allocated into core stats has climbed dramatically. Certain affixes, like life steal, have been cut while others, like Magic Find, are significantly less common. Drop rates and item availability have been increased to compensate; it's common to find legendaries and set items at lvl 70.
Unlimited, Shared Paragon Levels: Players who hit lvl 70 (or have previously hit 60) and continue playing will now keep their Paragon levels across all characters. Previously, each of your characters could be a Lvl 100 Paragon, but there was no sharing Paragon levels across characters. In addition, Paragon levels are no longer capped. Want to be a Lvl 320 Paragon? You can be.
There are a ton of other changes that don't fit neatly into bullet points. Crafting has been streamlined, there's a new time of socketable gem (Diamonds), and a new Mystic that can re-enchant items. There are some hardcore-specific items that will allow players to resurrect -- provided they've crafted, and are wearing, an expensive one-shot item that has to be re-crafted in order to function again. There's a hardcore unique that accelerates your town portal, reduces the damage you take while casting it, and makes it uninterruptable.