Internal stories for rpgs focus on the character's backstory and use this directly to create a story about that character in particular (or group of characters). Example: Recently a friend and I played Don't Rest Your Head a strange game about insomniacs who wake up to a nightmare city. The game's character creation strongly favors an internal story. The character sheet lets the character decide what just happened to them, what's there path, in addition to more typical questions like their overall character concept. This was somewhat refreshing and also strange because I couldn't prepare a lot for the game (I was GMing it). The player had decided his character was a sleazy Vegas guy who's just married a stripper who turned into a monster in front of him. From there we added that she'd ripped out his happiness (a hexagon jewel) and fled out of their hotel room into a nightmare world he'd never seen before. From what the player gave on his character sheet, how the game began was decided. We added a bit of extra motivation or plot goal (get happiness back), but the player gave the setting and primary antagonist, and from there had much control over how he went about this, and what could be found within the Mad City (this setting is a great combination of Neverwhere and Dark City). It was interesting because neither of us knew where this story would go, and compared to the more external based stories I'm used to telling, his character had great personal investment in this particular tale.
External stories in rpgs provide motivation that isn't dependent on any particular character or their background. The character's backstory may factor much less in this, however this has the advantage of allowing a group of characters to hold the same external goal and cooperate better. Example: In Changeling the Lost our motley is the official freehold detectives. Some of the characters were once policemen or investigators before this, but others held jobs unrelated to detective work. With the premise of being the detectives however it offered plausible motivation for a group of characters to work together to solve problems that they had nothing to do with the origination of. - The bulk of our stories were focused on single session mysteries.
In what I've said so far I believe I've touched on some pros and cons of both styles:
Character has great control over the story, and their past is very relevant to current events. This can create very character driven stories that are easy to make up on the spot and require less planning. NOTE: Because Internal Pros is shorter than External Pros please do not take it that there are less reasons to play a more internal focused game.
Story is limited to solving character's goals, and once this is finished the game is essentially over. This isn't necessarily bad, but it's not good if a game is meant to be self-sustaining for a long period of time. However, a story-driven player (I define as a player who doesn't play a rpg to "win" but to constantly be engaged with story and conflict) or they and the GM might constantly introduce new internal or external conflicts might become internal as time goes on.
It is much simpler to have a larger group of players use an external goal to rally under. The above example of Changeling provides an external goal, but some characters were made as detectives. This implies that a few of the characters have an internal goal to want to investigate and solve citywide puzzles, murders, et cetera. Even further, a player could create a character like Agent Mulder from The X Files: he is a special agent, but this isn't just a job to him. His sister was abducted by aliens and this has driven him his whole life. He attempts to solve all strange cases because they bring him further knowledge that could be truths of his sister's abduction. Having a similar goal in our chronicle of Changeling would take what the GM has specified as the external goal: You'll be the detectives of the freehold, and internalizes it so the character's primary motivation is also the external goal for the whole group. This is not necessary, but it offers a happy medium between external and internal motivation where a character is likely to be useful and driven when working with a group.
If a character isn't driven or interested in the external goals that the group or worse - they by themselves - are meant to be driven or interested in then alienation from the story is likely to occur. This is especially likely if the GM doesn't specify what genre of story the chronicle will focus on or tell other important details like mood or theme (if the GM has explicitly thought these out), or give some other indication to allow players to plan their characters with reasonable expectations.
Avoid the following mistake:
For awhile when I first started GMing I thought that it was better not to let anything out about a campaign or chronicle. The less players knew, the more in suspense I could keep them and the more surprised they'd be. From reading forums, I take this to be a common rookie mistake, but I'd like to state what was wrong with this: If you create a story that is intended as high mystery, low violence, and do not tell your players this, you could easily get a group of bruisers with no interest in solving a mystery so much as beating the crap out of whoever did it. In which case both players and GM become bored or blame the other for not going along with the game that they want. It's important to share if a game is going to be a mystery, political conspiracy, violent, et cetera. You can tell the players that they'll be solving a mystery without giving away the specific details of the mystery they'll be investigating. Writing that out right now, that reads as obvious advice, but I wish someone would have told me that.
Most games I've played or run have leaned much more on the external side. As I said before, I believe this is the simplest way to run a game when dealing with a large group, but simplest is not the same as the best. Several game books I've read recently have put forward the idea of having one character or another stand in the limelight and have their turn to be the protagonist. One game in particular, Spirit of the Century, utilizes a mechanic called "Aspects" some of which can be possible plot devices such as "Rival of (Insert NPC)". In such a case, if a player goes out of their way to name and create a rival, then there is very good reason for a GM to include that rival in a scene or build a whole campaign using that npc as an antagonist. As long as the GM doesn't alienate the rest of the group by doing this, a more personal, character driven game can be played. Also, as earlier stated, a happy medium between internal and external goals can be reached as long as the GM is willing to share key information and work with players. Hopefully this creates accurate expectations and characters that not only can meet the conflicts they face, but do so for meaningful reasons.