My second playthrough of Final Fantasy Type-0 ended yesterday. There are some differences compared to the first run:
-new missions are unlocked under the codename "Code:Crimson"; it lets you play alternative missions on the main path, just choose the second option
-you can unlock the bonus "Secret Ending" which is a massive Cosmic Retcon to the events of the game
-free time events you viewed on the previous run does no longer consume any time so you can finish all conversations

Story-wise, nothing changes. The bonus missions are just tougher than the default missions BUT feel completely disconnected from the main plot and add nothing new to the story or to character developement. As for the secret ending, it's pretty funny to watch Class Zero spend a more or less normal school life but it's too short and overflows with so much happiness that it feels unreal. But I guess they deserve a full-blown happy ending after the crap they had (not) been through

tldr; It is worth replaying FF Type-0 if you like additional missions, challenges and sidequests. If you expect more story or character development,then the amount the second playthrough adds probably won't satisfy you.