The story mode takes place in a theatre. You are acting out a performance at the theatre in front of an audience. It is a mystery game, similar to that of cases of Sherlock Holmes, and other detectives like that. SpongeBob plays the leading detective and the other characters play other parts in the game. While it seems quite simple, there is a much larger complicated system to each scene. There are three chapters overall and there are four to five scenes in each chapter.
Arcade mode is a mode full of mini-games. The mini-games are small challenges that you have already completed in the Story Mode.