From Diablo Wiki
- Campaign Mode or "Story Mode" is the other primary game mode, where players must progress through the acts in order, complete quests, talk to NPCs, watch cut scenes, etc. This was the only game mode previously found in games in the Diablo series.
Adventure Mode was partially known from datamining, but was first publicly displayed at Blizzcon 2013 where it was playable in the demo on the show floor, and drew very positive fan feedback. It is meant to serve as the new end game option for most players, and allows for very quick or very long game sessions, solo or with friends. It was very positively received by players at Blizzcon 2013 and is a much-anticipated feature.
Adventure Mode Features
There are several key features of Adventure Mode, accessible only in this game mode.
Open World: Adventure Mode is a no-story, "open world" style of game, where players can move freely between the acts and jump between any waypoints without even returning to town. There are no sub quests or story elements in Adventure Mode, and all boss dialogue, cut scenes, and event stages have been removed. All bosses are available and players can skip from area to area killing just the bosses, grind through all the levels in sequence, clear out just their few favorite levels, or any other play style that's appealing.
Flat Difficulty: The Normal/Nightmare/Hell/Inferno difficulty levels are not present in Adventure Mode. Instead the game employs a dynamic system to scale monsters and rewards to the character's level. Bosses gain additional Elite Affixes at set character levels. (Level 33 characters in the Blizzcon 2013 demo saw random bosses with two affixes, as would previously have been be seen in Nightmare.) It is not yet known how Adventure Mode will scale the difficulty for multiple characters of varying levels in the same game; by the character who creates it?
Bounties: There are 25 bounties available in each new game of Adventure Mode; 5 per act in each of the 5 acts. Bounties are usually missions to kill specific monsters, either randomly-occurring Purples or various quest bosses including Act Bosses. Other less-common bounties are awarded for completing an event or killing all the enemies on a specific level of a dungeon. Completing all the Bounties in a given act earns each character a random bonus reward which may include a legendary item and always includes a Rift Keystone.
Nephalem Rifts: These special dungeons are entered through yellow portals opened at a special obelisk in town. Opening a Nephalem Rift requires (and consumes) a Rift Keystone. Nephalem Rifts are fully-randomized dungeons of 1-10 levels, filled with random arrangements of monsters. Killing enough monsters in a Rift will summon the Rift Guardian, a randomly-selected boss with special powers and a nasty disposition.
No More Repetitious Story Mode Grinding: Aside from the new features, characters can play through Adventure Mode indefinitely; whatever the minimum requirement to begin AM, since the difficulty scales with the character, a player could level up from 20 or 30 or 40 to the maximum. All in Adventure Mode, without ever returning to Campaign Mode.
The requirements are still being debated internally. At most a new character would have to progress through all five Acts in Campaign Mode in order to access, but there is also the possibility that a character could start in Adventure Mode before that point. 
The issue here is whether characters would be able to start in Adventure Mode as soon as they completed each act, but with only access to that act via Adventure Mode. e.g. once you killed the Butcher in Act One, you could play Adventure Mode, but only Act One would be accessible. The alternative would be to make players complete all five acts and then start Adventure Mode with the entire game world open to them.
- Unlocking Adventure mode - CM, Blizzard Entertainment, 13/11/13