In the winter of 1625, Riordan O'Neill, a King's Musketeer, and Guillaume Sébastien, a physician, were, at the behest of the duke of Savoy, dispatched to Milan to seek the ransom of the Sabaudian vicomte de Praz-de-Lys, a captive of Don Alvaro de Salamanca, the Spanish governor of the Forte de Fuentes. Along the way, a French diplomat informed them of an important fact - should Don Alvaro, who to this point steadfastly refused to ransom his captive, die, the vicomte's parole would pass to the governor of Milan.
Mission-oriented advetures are a staple of roleplaying games; because they made for convenient tournament contests, many of the earliest roleplaying game adventure modules were of this sort, and as a result of their initial prevalence, the model became deeply ingrained in the expectations of many gamers. Rescue-the-hostage and retrieve-the-mcguffin were - and are - a staple of gaming action, so much so that when a poster at Big Purple referred to this as "trad," I had to agree, my own sandboxy proclivities notwithstanding.
I believe it was 3e D&D that referred to the 'trad' adventure model as 'linear adventures.' Linear adventures are the now-classic scene-scene-scene-climax, or event-event-event-destination, or encounter-encounter-encounter-boss fight, format found in most game books and modules. Despite the linear set-up, they may be experienced in a non-linear fashion, as when the adventurers succeed in bypassing one of the scenes through smart play or dumb luck, such as avoiding a bandit ambush.
Railroading, on the other hand, doesn't allow the adventurers to bypass the encounters the referee or adventure sets in front of them; attempting to do so results in the players being forced into facing the encounter, as when players' characters attempt to take the west road instead of the east road with the bandit ambush the referee has planned, and a massive lightning storm springs up that keeps the adventurers from going west.
Illusionism is railroading's kissin' cousin. In this instance, the players are free to go whatever direction they like, but no matter what they do, the referee springs his encounter on them anyway, as when the bandits attack the adventurers regardless of whether or not they take the east or west (or north or south) roads.
Unfortunately, some suggest that 'railroad' is somehow the opposite of 'sandbox,' ignoring or failing to understand that a referee can just as readily railroad the players and their characters in a sandbox as in more traditional adventures. Railroading and illusionism, are, to me, profoundly dysfunctional ways to play roleplaying games, and conflating all plot-driven adventures as railroads is a serious error.
Moreover, I don't think scene-scene-scene-climax is actually a good approach to defining linear adventure so much as it's the most common way it's presented in roleplaying games. A linear adventure is more about operational progress from a discrete starting point to achieve an explicit goal or end which is frequently external to the player characters, that is, the goal is imposed upon them by someone else. It is often structured as scene-scene-scene-climax in order to ape a story's rising and falling action, conflicts and crises, and conclusion; the problem is that much of the advice to referees for running linear adventures expects or encourages railroading or illusionism in order to maintain that structure. This is poor adventure design, of course, but it doesn't inherently need to be that way.
Rather than structuring a linear adventure as scene-scene-scene-climax, I may prepare a slew of options for how the npcs react depending on what the adventurers do, like in "The Lady of La Rochelle" for Flashing Blades, or set the actions of the npcs against a timeline in which the antagonists are acting and the adventurers are free to respond, like in Operation: Ace of Clubs for Top Secret, or as a series of events to which the adventurers may or may not respond, like in Burned Bush Wells for Boot Hill.
Burned Bush Wells is an adventure which takes place in the eponymous town during a hard winter. There is a conflict between two factions, a powerful businessman and his crony marshal and the other business owners of the town. Sequential events in the adventure detail the steps taken by each faction to overcome the other, concluding with one faction or the other winning, depending on the involvement and actions - if any - of the adventurers.
BBW can be viewed as scene-scene-scene-conclusion, but it can also be viewed as a timeline which exists independently of the adventurers; the adventure makes only very general assumptions about how the adventurers will respond, if indeed they respond at all, to the sequence of events. It's entirely possible for the adventurers to bypass all of the events, side with either faction, or progress from starting event to ending event in order.
The events of the adventure unfold linearly - a thief, helping the doc, a shotgun wedding, and so on - moving toward a conclusion. The events are even dropped at the player characters' feet. But what BBW doesn't do is presume the adventurers's responses to the events, and it loads up the referee with enough information to wing it if the adventurers try something completely different, and it even gives advice to the referee on how the events change based on adventurer action and inaction during previous events.
When you get away from trying to structure the actions of the player characters as part of a story, then linear adventures can be constructed in such a way that they still lead from A to E but provide for meaningful player choices instead of ham-fisted transitions between scenes or encounters.
Linear adventures make sense in a sandbox when the player characters submit themselves to some authority - they join the army, they are inducted into an order, they buy an office, they are granted a title, &c. Requiring adventurers to fulfill their responsibilities gives meaning to accepting titles, offices, ranks, and so forth, and linear adventures arising out of those responsiblities may be one logical way for those duties to present themselves to, and impose themselves upon, the player characters. Unfortunately, linear adventures - specifically badly conceived and presented linear adventures - have poisoned the well to some degree, leading some gamers to dismiss them out of hand and even to claim that any attempt to control the actions of the adventurers - especially in a sandbox game-world - is railroading. This does not need to be so, but referees should also take care not make the fear become a reality when introducing linear adventures into their sandboxes.