There are a lot of differences, but one of the biggest is this: DEs don't just fill the role of rifts, but also replaces traditional quests. In a game like RIFT, the bulk of the content is solo-oriented questing, and then on occasion a rift will pop up. Once that rift is done, people retreat back to their quest lines. In GW2, DEs make up the bulk of PvE content: there are over 1500 scattered throughout the world. And because there's no need to form parties - there's no allied targeting, so no need to see players' unit frames; there's no mob tagging, so this "who can tag the mob first?" mini-game is gone; most every support skill works with or without a group; etc. - people naturally cooperate with one another. People won't by default disperse back to their individual corners once a DE is done, either, because there's a constant flow of DEs taking place and they're often chained to one another; rather, over time ad hoc groups will form in this organic way.
GW2 is built from the ground up, whether it's in terms of combat mechanics or the PvE content, to be a social game. That's the main reason DEs will be more successful than rifts. Of course, DEs are also far more varied, have far more interesting effects on the game world, scale very intelligently, and so on. =P