When reading a book-series or popular movie that is sure to see some sequels, you´ll most of the time get a fine, continuous piece of story. Even though each book or movie is fine by itself, the follow-up will make it clear to the audience that what happened in the past is relevant for the present time. That doesn´t hold true for video games.
Most video games are fully enclosed products, standing on their own. That is fine as long as a publisher doesn´t decide to turn that single game into a recurring franchise. Even though nowadays video games have become big, epic and, unfortunately, very cinematic experiences, they don´t really try to tell ongoing stories. Instead, they welcome you with the same main character and have you start a whole new mission. What happened in the previous game is but a page out of a book, it bears no more relevancy to the new plot. Part of that is also that video game makers aren´t daring enough with changing the main character. Having him develop both in terms of looks and behavior depending on his past adventures. It´s always the very same dude/girl. Be it Uncharted or Tomb Raider, Mario or a hundred else franchises – never do you get to know the bigger picture, never happens something of permanent relevance.
There are examples of games that successfully achieve such relevant continuity throughout their various installments, but these are rare and even then often rather shallow. Most prominent example would be the Mass Effect-franchise. Of course, being a western-rpg it is easier to include permanent developments since it is the players decision of what to do or not to do. Also, Bioware has the franchise planned as a trilogy from the beginning, so there is no uncertainty about how to proceed with what you have. Two other good examples, surprisingly, come from Nintendo. As much as story is not a big part of Nintendo-games, both Metroid and Zelda succeeded in creating a big overarching continuity that bears relevance to future and past games. Be it Samus destroying the home planet of the Metroids, later on fighting Mother Brain and then herself aka SA-X – each game takes its prequels into consideration. On a less pronounced basis, the same is true for The Legend of Zelda. This series´ fans are famous for their timeline-talks, and while I personally wouldn´t take pre-OoT-Zeldas into such talk, Ocarina of Time definitely started a continuous story that saw relevance even in the latest console-title Twilight Princess, showing scenes of Ganondorf and what happened to him in the aftermath of OoT. And then there´s the Triforce-mythology that overarchs each and every series-entry. Even the upcoming Skyward Sword seems to relevantly further the whole franchise´s continuity by showing fans the origins of the Master Sword. One very popular Konami-franchise also is known for its continuous story: Metal Gear Solid. Hardly a surprise, when this franchise´s focus is story foremost, and complaints about its individual installments´ uber-long cutscenes are common occurrence. You could also name the Kingdom Hearts-franchise, though I wouldn´t include it here. Other than the other mentions, KH feels like Square Enix is making up sh*t for each new game, further fucking up the whole series. And let´s not forget Assassin´s Creed, where Ubisoft just like that makes up another game only because of the franchises success. Killing an originally interesting story by thinning it out by filler. Oh well.
I´d really love to see video games being treated more like books/movies in that regard that their stories and characters keep developing instead of just creating new missions for a never-changing hero. And just in case someone wants to call me out for mentioning the Mario-series in a text about story: The Paper Mario-games have great, franchise-fitting stories. Continuity would work there just as well and maybe finally give Luigi the development he deserves. Luigi > Mario.