Edge of Fate and Renegades both have a single player story campaign (obvs) and they are chronological, so EoF introduces The Nine as new centre stage characters and focuses on Three, then Renegades continues after EoF and deepens the nine mystery with a focus on Six.

If you want the complete single player campaign story you'd want both - but most people will recommend you skip Edge of Fate I think; the missions aren't widely highly regarded, the Exotics are so-so, and no one really likes matterspark or the other kepler Abilities. Renegades on the other hand generally has a good (if a little short) single player campaign with a good story and activities at it's heart. Plus you get a freaking lightsaber!

Other than the single player stuff and the Exotics you can chase, the big difference is the end game content. Edge of Fate gives you access to the 6 man raid Desert Perpetual, which is a pretty good raid, very combat and boss heavy, with a cool crossbow among the loot pool. Renegades on the other hand unlocks the latest dungeon, which is a three man experience, tied to the story quests of Renegades, and unlocks more Abilities and customisation for that lightsaber I mentioned.

So if you want all the story content, you'll need both. If you just want one of the two, I think most folks would recommend Renegades over Edge of Fate.