These missions are the subject of a dev blog from EVE developer CCP Jasonitas, who explains what sets Epic Mission Arcs apart from standard quests: "Epic arcs are a different breed, a different style of mission altogether, and as such require the player to approach them from new angles and perspectives [...] You are playing an immersive story, with dramatic arcs, unique characters, and an ebb and flow of dramatic events", including story branching.
A major aspect of Epic Mission Arcs is that they take advantage of the size of EVE's sprawling galaxy. With standard mission running, players tend to find an NPC agent to stick with and then only venture a few solar systems in any direction over the course of their PvE gameplay.
This is not the case with Epic Mission Arcs. CCP Jasonitas says, "Pack your bags. Be expected to travel through different regions, constellations, and systems. New Eden is a big place: This is your chance to explore part of the universe and to poke at EVE's hidden corners, the nooks and crannies of deep space."
CCP has also added more texture to these stories with a greater levels of detail than in standard missions, some of which is hidden.
Jasonitas says: "Flavor is everywhere, tucked away in bits and pieces strewn throughout EVE. Thus, we encourage you to take in the story and find the flavor wherever you can: NPC descriptions, character bios, item descriptions, storyline items, the works. We placed bits of story everywhere you could think of, and everywhere we could. Don't be afraid to look for those little pieces that could enhance the world you've entered. Sometimes they might just be for show - other times, they could have a deeper purpose."
As with playing as a capsuleer in EVE Online in general, the Epic Mission Arcs maintain that sense of moral ambiguity that's woven throughout the game's setting of New Eden. The things you do and the choices you make won't necessarily be 'heroic'. Jasonitas writes, "EVE is a dark place. Yes, these epic arcs may be about larger-than-life stories in which you play the part of the protagonist [...] But New Eden isn't a black-and-white world. A hero to one group is a villain to another. Sometimes, you'll have to make tough choices. New Eden has just as many anti-heroes as heroes. Remember that, as you destroy that fleet of battleships or plant that bomb to destroy a station full of civilians."
These stories play out over the course of many interconnected missions and there are rewards (hopefully) commensurate with the risks taken and effort put in, though some players already disagree on that point. These rewards can be large sums of ISK, of course, but it can also be other unique rewards in the forms of modules and equipment. Each epic arc has at least one unique reward. Jasonitas says, "Some may have more, based on the ending you choose or the path you take. You may have to do some nefarious things in order to get the sweet loot, but that's all part of the epic arcs experience."
For a synopsis of the storylines which run through each of the Epic Mission Arcs, see "Epic Is As Epic Does" from CCP Jasonitas.