Reason for Naomi's Death: Nanomachines subplot pulled out of ass at the last second to reveal that Naomi has kept cancer at bay for years. Otacon has known Naomi for weeks at this point, if that long. Scene ends with Naomi talking to a small robot with Otacon bawling uncontrollably while Naomi deactivates both her nanomachines and Vamp's (Vamp's relation to Naomi is never expanded upon much either). There is no character development. There is no attempted moral to the story. It's drama for the sake of drama.
Reason for Emma's Death: Stabbed by villain character. Root cause for consistently recurring sibling friction is revealed to be dark family history with parents previously touched upon in scenes with Raiden. Otacon has known Emma since adolescence. Death scene ends with Otacon granting Emma's final wish by calling her Emma rather than E.E., a gesture illustrating a development in their relationship. The parrot established earlier in the game says "Hal....I miss you" revealing that a connection between characters was always there. After crying in a second scene, Otacon resolves that you can't expect love to just find you but you must go looking for it (the moral of the story) and goes on to save the hostages, later rejoining Raiden as a regular codec contact.
Both are melodramatic, but one is clearly written worse than the other. Emma is mentioned early on in the Tanker and as Raiden you get to know both her character and relation to Otacon pretty well. She's clumsy, a little catty, and a complete scaredy cat. You can infer she's a "bit of an oddball" just like her brother. She claims to hate Otacon but obviously still has a soft spot for him. If you asked me what Naomi's defining character traits are I would have to respond with "scientist" and "cleavage". Her relation to Otacon is "they are both nerds who had sex in a helicopter".
Sunny of all people had a more clear cut relation to Naomi with hints that for a short time Naomi fulfilled a motherly role.
Snake has quite an obvious history with Naomi as well. These two characters could've shown more reaction to her death, but Otacon's reaction was just there to draw another pointless fan service parallel to Sniper Wolf.