Story has always been an important part in some games. A lot of games still have story of some sort even if they have no cutscenes or dialogue. Some games simply feature a "story" button featured in the main menu or something of the sort (though that's usually for flash games or lower budget quick games). For titles with larger budgets they usually have either cutscenes with dialogue & voice acting, cutscenes with only text dialogue, or the rare no dialogue and no text ones.
These aren't too common and it's very important for these kinds of cutscenes to convey all the emotion and the messages they need with only character motions. It becomes very hard to pull this off for certain kinds of games, such as RPGs which usually have the largest of backstories. However there have been some very notable successes that have proven that you don't always need talking and text to convey what's going on in a scene.
The best example I have experienced of this is the Lego series games, such as Lego Star Wars, Lego Batman, Lego Indiana Jones, etc. In these games, the Lego characters do not talk at all but they have to re-enact scenes from the movies or settings they are in as though they were mimes. They are able to provide and make the emotions obviously in certain scenes such as worry, happiness, laughter, while still trying to stick to the character they are playing. They also have to use a lot of exaggerated gestures to convey more emotion as well.
Lego Star Wars 2 - Cutscenes
The best way to relate how these are successful is black & white films from long ago, the ones that featured no sound at all. This is pretty much how these cutscenes go. In order for the Lego people to properly convey the story to the player, they must adapt to lack of sound pretty much (at least lack of dialogue). Movements must be over exaggerated , expressions must be exagerated as well to easily convey how the Lego person is feeling. Where the camera is looking is even more important as it must make sure to focus on whats important in telling the player what they need about the story and to show the character expressions at the right time.
The Artist - A recent movie done like the silent films from long ago
A no text & no dialogue cutscene is entirely possible and can be really well done and successful. The saying "actions speak louder than words" definetely rings true here, as the case of Lego Star Wars has shown just how much can be displayed. However this isn't easy to get right. If you can't get the characters to display the right expressions, it can come out as half-assed cutscenes that don't have much merit.
Also trying to imagine it in a game with more serious story is difficult. Though movement and facial expressions convey a LOT of emotion, speech conveys another layer that adds to a scene and the world. It's hard to imagine a large Final Fantasy like world being able to convey everything that's needed without dialogue and still be fully enjoyable and as fleshed out as it could be with dialogue.
Dialogue - With Text
We've seen these types of games for eons, since the days of Zork and with advent of visuals, in the form of cutscenes in RPGs like Chrono Trigger, Dragon Quest, and even Zelda. Games that feature only dialogue with text read on screen are still common, however in the triple A game area, it's becoming fewer and fewer. Most games that don't convey much story can just stick with this, but those that do need to convey a story have gone to sticking with actual speech instead.
However there are some notable exceptions that show that so much expression of the world can still be displayed without the need for characters talking. One of the most recent examples I have played is The Legend of Zelda : Skyward Sword. Characters are extremely expressive in terms of animation and facial expressions. They convey absolutely everything necessary to go along with the dialogue you're reading they are saying. You can easily match the two together because of how well the emotion is conveyed.
Legend of Zelda : Skyward Sword - Has some of the most expression I have seen in characters in a while
This is in fact very similar to No text & no dialogue, in how much importance there is on character expression at least for Skyward Sword. Though there isn't as much reliance put onto the expression of the characters with text, it's still a very important aspect if you want to get the feel of a cutscene right. 2D games can still manage to get away with only text but are they really as good as games that provide both text and some meaningful expressions? This is up for debate obviously.
Final Fantasy VI - Had plenty of dialogue and no speech
The thing about dialogue but only via text is that, it's much like reading a book in that sense. When you read a character's speech, you can imagine what their voice might sound like in saying it. You get to embody the character with however you want that character to sound. If I want Zelda's Father in Skyward Sword to sound like Gandalf, then I can do that. This applies to both 2D and 3D games, perhaps even more so for 2D games as you don't always get character expressions on screen too.
Dialogue - With Speech
Dialogue with speech can be considered to be the whole package, but really it's simply a different art form. You now have the final layer of audio to go along with movement and expression, speech from the characters. In most triple A titles, this is the choice they go with for cutscenes and with good reason. Movies nowadays are no longer soundless, they come with people talking, music cues, etc. Most people are used to these kinds of cutscenes and enjoy them more so.
However these aren't always the best choice because it relies on the talent of your voice actors. In fact it can be detrimental if done wrong. A terrible voice actor can take you out of the experience with their dull and flat lines. Dialogue that's in written form, when conveyed in speech can actually sound terrible, but seem fine on paper. You have to rely on people saying the lines in a way that sounds good and fits the character, while making sure their lines sound naturally. Take for example some of the games that get dubbed from Japan, some of them are terrible.
Trouble Witches Neo - Terrible line delivery
Those are the downsides, but the upsides if you can get it right, breathes new life into the character. Much like you can get attached to a character in a movie, you can get a attached to a character in a game if their voice conveys their emotion and personality in such a way that appeals to you. Of course you can still get attached when they can't speak, but if the voice really fits the character, then it makes it hard to imagine the character not speaking that way. For example once I heard the voices of the characters in the movie for Lord of the Rings, whenever I read the book I always hear their voices now. So strong the voice acting can be, that these actors and actresses behind the voices can sometimes be permanently imagined as that character, such as Nolan North for Nathan Drake in Uncharted.
Uncharted 3 - A series known for it's excellent dialogue and line delivery
These kinds of cutscenes can also be more immersive. Take for example some of the cutscenes in Modern Warfare 1. The helicopter scene when the nuke goes off is great because the dialogue brings you into the scene. Imagine the build up to that moment with just text, I don't believe it would be quite as impactful. It would have been slower and the stress of the situation wouldn't be quite as obvious to a lot of people. It would also have to make some changes into how it was shot to make it more impactful without any dialogue if you tried that method. I wouldn't be able to take it quite as seriously if I just saw a bunch of overexagerated expressions.
Call of Duty 4 - A shocking moment helped conveyed by dialogue
I can't say one is better than the other because that's not true. Take these as artforms, different ways to convey a message, in this case a cutscene. I like to imagine Action only as the silent films from ages ago, text only as reading a book (or watching a subtitled movie), and speech as a hollywood movie. These all appeal to different audiences and provide different experiences for the viewer. Sometimes they aren't always done well and can fall flat on their faces, but they can each be entertaining in their own right as long as they are done right.