Bach himself made this distinction when he wrote to Georg Michael Telemann in 1767 to clarify his duties in Hamburg: "are [Passions] presented in the historic and old manner with the Evangelist and other persons, or is it arranged in the manner of an oratorio with reflections, as is the case in Ramler's oratorio [Der Tod Jesu, arguably the most famous setting of this text is by Carl Heinrich Graun]?
The Gospel text to be used was chosen on a rotating cycle, as was the Hamburg tradition established in the late 17th century, in the order Matthew, Mark, Luke, and John.
As they were performed in a regular Sunday service (not at a separate Vespers, as was the custom in Leipzig), Bach modeled his Passions on those of Telemann: they were roughly an hour long, and began in the Garden of Gethsemane and ended with the death of Jesus, rather than telling the contextualizing details as well.
The biblical text was set in recitative and assigned to the appropriate characters (individual singers taking the roles of the Evangelist, Jesus, Peter, and so on).
Reflective chorales and arias were inserted at predefined points in the narrative, providing commentary on the Passion events.