Scenarios you can say are... quests. Missions. It can be anything from dealing with a Disaster to making your way through a Demon Realm. There's often a time limit to it, a set of rewards you can obtain by completing... and sometimes heavy consequences if you fail, such as death.

[ not always fun and games. ] Story weapons are often things tied to a story. A bit different from Star Relics, like my Flames of Retribution. It normally depends on the story. ...It's a lot to explain.