Co-op missions are missions where you have to have a certain number of players in the team to finish the mission, you need to cooperate with each other, too.
"Party" missions are always on time.
You need 4 people to do "Her Secret Mission" in the GUARDIANS Ohtoku Branch.
Same goes for Bruces Dungeon located in the GUARDIANS Moatoob Branch.
As for other missions.. (which are "Free" going from Amesani)
You need at least 2 people to finish the mission Cost of Research which you can do backwards from the Kugo: Hot Springs.
And I think you need a certain number to do Stolen Weapons in Moatoob, too.
Well, that's what I know of so far, dunno about other missions.
Well, basically, they are missions you can only finish/complete with at least 2 people or more.
Else I don't think there are other big differences..~