A key element for writing a convincing character is to convey the idea that their universe is real to them. Their decisions have weight and impact - their losses are tragic, their victories are triumphant, the sensations they feel are connected to what's happening to them and around them, and so on. This comes from a direct, fully-sensory link to the universe: what's happening to the character is, well, what's happening to the character. If they get slashed with a sword, then all the associated baggage of being slashed with a sword should occur to them unless there is a reason not to. The drama of the story hinges on the idea of the character actually caring about what happens to them - if they aren't interested, then why should the audience be?
The player, on the other hand, cannot be as invested, due to the combination of lacking sensory detail and "real-life" consequences. A player, like any member of an audience, can be deeply invested, and they can roleplay, but it's never going to be the same for them. There are a bunch of ways to try to make the player care about what happens - likable characters, events with consequences, all the aspects of believability discussed in previous articles - but ultimately "the game" is not reality.
However, one interesting correlation about this is that dissension usually arises in cases where a character cares and the player who controls them doesn't. In Grand Theft Auto 4, Nico Bellic switches from "concerned individual with friends and comrades that he cares about" to "guy who just ran over a sidewalk full of people for no apparent reason". In other sandbox games, like Saints Row 2, the sociopathic nature of the protagonist matches the player's usual actions much more closely. Of course, this isn't absolute - there are plenty of people who play games in a "character appropriate" way, but the issue is that it's not important or notable to the story. There's "the character" and there's "the player", and ne'er the two shall meet.
There are a couple different ways to approach this situation, going from most-divisive to most-immersive.
Independent Character / Controlling Player
This is the scenario described above. The character is their own person, and when not being controlled by the player they make decisions as they see fit. When the player is in control, they may act totally contrary to the way they see themselves and choose to act. The player is the character, for a given definition of "is", but the character is also their own person. One of the issues of believability regarding this setup is the fact that there's no repercussions for the player-controlled sections of the game. Perhaps the character will quip or remark in response to a situation, but there's no real commentary on what's happening. The rules tend to work differently, as well - the usual mix of "cutscene bullets kill you in one hit" and "dying is a big deal" and other gameplay/story segregation concepts.
An approach on this that I feel would be interesting is the idea of this divide existing in-universe as well as in the meta-sense. That is, the player represents a spirit or entity with no connection to their realm. Occasionally, the character allows the player to control their body and imbue it with superior speed, strength, etc. This addresses the issue of motivation (the player acts as a spirit that acts according to their own whims rather than consequences) as well as justifying the difference in action. Perhaps the player and character could even come into conflict depending on how their viewpoints differ. "Deadly Premonition" plays with this concept, but does not create an adversarial relationship out of it. It is less "direct possession" and more of an advisory role.
Dynamic Character / Choice-Making Player
Not all games make a character "independent". Some games, such as the majority of Western RPGs, make characters variable in differing ways. These are limited by the reactions that the developers include, as well as any existing meta-systems like a good/evil meter. In short, though, it seems like having decision-making aspects should make the game more believable - after all, now the player is invested in the character, and the character isn't simply an immobile, unresponsive part of a linear story.
However, I'd say that there is a downside to this concept, and it is that "there is no longer a protagonist who is part of the universe". The protagonist is now motivated by the player's decisions, and the player remains unconnected to the universe in terms of consequences and senses. It's more like the character is now an avatar for the player; even if the player is establishing roles and making decisions based on their conception of the character, their motivations are still based on whims, rather than consequences. Death is "Oh, I lost" and not, you know, death. Again, there's nothing you can really do about that - it's just that extending choice to the player character includes them in that disconnected state.
Essentially, the "RPG character" is a strange hybrid. On the one hand, they exist in the universe and are supposed to have all the sensory/consequence issues that real people have. On the other hand, their choices are made by a person who doesn't have any of those. This means that their choices aren't going to be made for the same reasons that real choices would be made except as a coincidental bit of roleplaying. Proper immersion can aid in the player making decisions that the character would be motivated to make, but this is by creating sympathy or evoking similar emotions. It's a complicated scenario, but "making the player care as though the world was real" is the goal of a lot of RPGs.
I should note that, theoretically speaking, there are two kinds of DC/CMP setups:
1) The character exists as their own person, but the player influences the kind of person they are (think Shepard from Mass Effect). This is often used to try and make up for the limitations of video games: you can't really make an "open-ended" protagonist, so the next best thing is to make a linear one with branching paths. You have to be Shepard, but you can choose how Shepard acts and what he or she looks like. You can't choose their voice though. The voice is there forever.
2) The character is created by the player (think any pen-and-paper RPG). This allows for the player to adopt a far broader range of roles even within the confines of the setting, but is difficult to execute in a game that relies on premade dialogue. In a P&P game, the "dialogue" is naturally generated, so it's okay to do that. In a video game, it has to be written out beforehand (with some room for dynamic generation), so a few "personalities" develop (Good Protagonist, Evil Protagonist, etc) which is the most likely cause for the first category's existence.
"The Player As A Character"
Similar to the scenario I described above where the player is an advisor or spirit, some games go with the idea that the player and their interface directly represents a character in its own right. For example, in "MechCommander", the game display and HUD and so on are explained as being the MechCommander interface. The player is a Mech Commander, and the "game" is the program they're using to command their mechs. This can be seen in the opening movie, where the (non-player) commander uses an interface similar to the game's interface to direct his units. The hacking-based game "Uplink" is another example: the interface is meant to be directly equivalent to an in-universe program. Even fantasy games can get in on it - in "Dungeon Keeper" and "Black and White", the player represents an ethereal spirit disconnected from the "real world" but able to influence it in various ways. When the player is "damaged", what's being damaged is their connection to the world.
In terms of immersion, this can be the best concept (if properly explained). There's as few unexplained game things between "the player" and "the player's role" as possible. The lack of connection might even be noted - what risk is there to the commanding officer throwing their soldiers' lives into combat? The relationship between a player and their troops might well be the same as the in-universe relationship between a commander and their subordinates. Like in previous examples, they may care and take things seriously, or they may not - it's all up to their whim. Unlike previous examples, this is something that makes sense in-universe, because now there is a reason for all the disconnected elements. Of course, there are still limitations on content and dynamic generation, but in general the link between the player and their role is much stronger.
So, To Sum Up:
1) The character cares about things in-universe because they're happening to the character. The player cares about things in-universe in a detached or isolated way - at best sympathetic, at worst sociopathic.
2) This divide means that players and characters make decisions for different reasons - even if both have a reason to do things in "the most optimal way", the different factors mean that "optimal" results in a different set of choices.
3) One way around this is to make the character the same as the player, sharing the same limitations on sensory information and the same disconnect from events. This explains the player's decision-making process in-universe, because they're a powerful, untouchable being whose whims determine their course of action.