Severed is single-player, Awakened is co-op.
Severed is GREAT as a follow-up to the severely underrated Dead Space: Extraction. ABSOLUTELY play Extraction before playing Severed. It follows two of the main characters of that game.
Awakened is arguably the best content of Dead Space 3 (not saying much), and is probably the most "Dead Space"-esque content you can find in that game. It skews more towards the horror, psychological hallucinations, and grim tone of the original two games, and it ends on a massive, dark cliffhanger that is - for all intents and purposes - the final bit of Dead Space content many of us ever expect we'll receive.