Rift is, in many ways, an homage to World of Warcraft. There's two factions, the UI and interface is very similar, heck, the rogues even use the same 5-combo points and energy system as their WoW brethren.

The most obvious difference is Rift's Soul system. WoW has 10 classes, each with 3 talent trees each, whereas Rift has 4 Classes, with 8.5 talent trees each (8 full talent trees, or "souls" as they're called, plus a class-specific PvP sub-spec). While in WoW, each class has a core bunch of abilities that they learn regardless of spec, Rift's abilities are *entirely* dependent on which soul tree you select. You pick up to 3 souls at once, and can spend as little or as many points in all of them as you wish.

The other major difference is the titular "Rifts", which are open world PvE objectives that dynamically appear (in both strength and number) depending on the active players in the zone. Rifts are an alternative to questing or grinding, and provide comparable gear upgrades at the planar vendors.

Like WoW, Rift has battlegrounds, instances, heroic instances, and raids. You can queue for the battlegrounds from anywhere (called "Warfronts" in Rift), and like WoW, Rift has a Dungeon Finder tool that lets you queue for dungeons, raids, and group quests.

There are undoubtedly more differences between the games (Rift appears to be much larger and harder to navigate; "flightpaths" aren't nearly as numerous as in WoW), but the Soul-system and the Rifts are definitely the two biggest.