Spurred on by the recent posts on the subject, I wanted to give my own opinion on this matter.
A little background: I'm the kind of person who plays all sorts of game. I've gone from playing competitive Counter-Strike, to playing obscure VNs, and almost everything in-between. I enjoy both games that focus on gameplay as well as story.
Games tend to either focus on story or gameplay, but there are some that do both well and a final "bonus" category I'll touch on at the end.
The Gameplay Game
Notable Example: Mario
When it comes to games that focus almost solely on gameplay, Nintendo is almost certainly the first company that would come to mind. I've never heard anyone ever say they play Mario for the story, which makes sense because there essentially isn't one. It's the gameplay that gets you coming back. The game is fun enough within itself that it doesn't need any other reason to get you to keep playing. These kinds of games are fairly autotelic, they're an ends in themselves. You don't have any other exterior motivation that drives you to play them, but rather it's a rewarding endeavour within itself.
The Story Game
Notable Example: The Walking Dead by Telltale Games
These kinds of games are the exact opposite of a gameplay game. If you stripped the story from these games, there would be absolutely no point in playing them. At the same time, for many of these games, you could strip the gameplay from them, and they really wouldn't suffer that much. In some cases, these games may be better off without their gameplay elements. The gameplay is the guide that brings along the story, and if done well, promotes these kinds of games into the next category. But, games with really good story can very well carry poor/non-existant gameplay. No one would bother to play a game like The Walking Dead if the story was bad. It's only because of the story, that the game is critically acclaimed.
Another example would be Planescape: Torment.
This game has incredible story, with rich themes and meaning. All of the writing in the game is top notch. However, if you played it today, the gameplay is abysmal, it's very confusing, especially if you're unfamiliar with Dungeons and Dragons rulesets, and interfacing with the game is an absolute chore. Yet, it absolutely needs to be a game, as much of the exploration and discovery is a requirement for the awe that you may experience when you play the game. If the game were remade today with a more modern, streamlined interface, I would recommend it in a heartbeat. However, I think the majority of people won't be able to get past its antiquated gameplay, and thus miss out on the story, which is a shame.
The Integrated Game
Notable Example: Persona 4
These are the games that seem to do everything right. The gameplay is fun and entertaining, but it also has an engaging story to boot. If you manage to make a game that does both of these elements well, you're certainly on your way to a successful game franchise.
Another thing that these games do that only video games can do is integrate its story with its gameplay. Persona 4 is a great example of this.
PERSONA 4 SPOILERS START HERE:
One of the main themes within in Persona 4 is how you need to search out for the truth, and not to be persuaded by simple or convenient answers no matter how well they seem to be the answer. The Investigation squad goes through many iterations of who they think the murderer is, but refuse to be persuaded, regardless of how obvious certain suspects may seem. However as a player, you have some impact on how far you will go out to "reach out for the truth" (which of course is the main battle theme for Persona 4). For example, if you push Namatame into the TV, and cut your search short you get the bad ending because you didn't search deep enough for the truth. This still counts as a proper ending of sorts, as you still can go and start a NG+ off this ending, although that's probably not recommended.
However the real amazing part about this comes at the end of the game. Once Adachi has been jailed, and MC is about to leave to go back to him original home, the game gives you the option of making sure you're finished with everything in Inaba. I'm certain for at least a good section of players, they don't really think about this option, and say yes, and head towards the normal ending for the game. That's all good, but, for the player that thinks a bit more critically, there are still unanswered questions. Who/what created the TV world? How did MC, Adachi, etc get their powers? How did this all really begin?
For people who have still considered these questions, and really wish to seek the truth, you can say no, and find out about what really what caused this entire situation. The entire section with Adachi and Ameno-sagiri was all really a ploy by Atlus to make you think everything has been settled, when it really hasn't. But they don't just leave you hanging, they're kind enough to leave hints. Why do you think you level up the Judgement Social Link to max social link right at the end of the game when there supposedly isn't any more fighting to be done? Why would they make a final persona for Judgement (Lucifer) when you supposedly are finished with the game? That's something people might think about, but not exactly piece together right away.
Only once you, yourself, the player, have resolved to find out the real truth behind this game's story, can you unlock the true ending. Only if you've really taken this game's message about searching for the truth to heart can the true ending be revealed to you (unless of course you read a guide, or just choose no because of curiosity etc).
However, trying to make a game like this carries a certain kind of risk. If you try to do both, but only end up decent in both categories, the game suffers from seeming mediocre. A recent example of this is Child of Light.
The game looks stunning; however it doesn't have either the gameplay or story to back it up. The gameplay is fairly simplistic once you've figured out the groove of things, and doesn't really become much deeper as the game goes on. It has the basis for great gameplay, but it doesn't build upon it, which makes things feel repetitive as the game goes on.
Likewise, the story suffers from a similar problem. It tries to go for a rhyming scheme where everyone rhymes all the time. It's ambitious and if pulled off, would be quite an amazing feat. However, it doesn't pull it off, and comes off as kind of tacky.
I really wanted to like Child of Light, but in the end, these problems stop it from being the game it could have been. In these cases, the game may have benefited from either focusing on the gameplay or the story.
Then there are the games that give you the one-two punch. Came for the gameplay, stayed for the story (could be the other way around, but this is probably the more common route). An example of this is Dark Souls
Dark Souls is infamous for its difficulty and its engaging gameplay. Players masochistically fight their way through each of the bosses dying countless times for the glory that they receive at the end. At first glance, Dark Souls is a textbook example of a gameplay game. Ask a person playing Dark Souls for the first time, and they'll almost certainly tell you they have no idea of overarching plot of the story is, or why they're doing what they're doing (which Dark Souls uses as another plot point, but I'm not going to write another chunk describing that, this article's long enough as is...).
After completing the game once, many players will put the game down, and they file it as a game they enjoyed for the gameplay. But people that continue to play the game will get more engrossed, and begin to wonder about the world that is Lordran. If you go through and read the item descriptions in Dark Souls, you'll find there's a whole lot going on in the world, and the lore is incredibly deep. If you think about why bosses/items are where they are, in conjunction with the descriptions, and talking with NPCs, you'll find there's actually a lot of very interesting stories going on that are incredibly easy to miss. Eventually some people may come to enjoy the stories within Dark Souls just as much as they enjoy the actual gameplay.
The "Your Story" Game
Notable Example: Minecraft
Finally, we have the outlier of the group. These are the kinds of games in which YOU are the one who carves your own story. These games tend to be sandboxes with no story to guide you whatsoever, but that in turn puts you at the helm of driving the story.
There's no one guiding you or directing you to do things, you do whatever it is that you want. This freedom means that the things that you do are directly attributed to YOU. You didn't really kill the dragon in that game; you battled a boss, and triggered a cutscene in order for that boss to die. You're not the first to do that, and you certainly won't be the last. However, in a game like Minecraft, you're actions are attributed to you. If you cut down that tree, that's something you did. Maybe someone else also would cut down that tree, maybe they wondered somewhere else. It's a simple example, but it's effects spread far and wide. Maybe in your story you RECREATE THE ENTIRE COUNTRY OF DENMARK, maybe in another's you DESTROY IT. But those stories depend on you. You're the X-Factor within these stories, and without you they wouldn't exist. In these kinds of games, you write the story, and they solely depend on you. It's why Minecraft and Minecraft videos are so popular, because naturally like to hear the adventures and stories of others. Search Youtube right now for Minecraft, and you'll find tons of videos of people detailing their adventures of their Minecraft characters. It's because these stories are unique, and that no one will 100% recreate your adventure one of the reasons that Minecraft has become such a phenomenon.
It's also by this reason that many competitive multiplayer games are so successful. Each game is its own story that you can retell and amaze others. Using Dota 2 as an example, there are a lot of stories there that any veteran Dota 2 player will be able to tell you. The Rosh Bait by NoTidehunter at Dreamhack had people going nuts, just like when Na'Vi pulled off "The Play" on IG, or when they fountain hooked TongFu to beat them at TI3, or when kyxy denied the Aegis which many would consider the misplay the cost them TI3.
But it's not just the pros, I'm sure we all have our own personal stories within games. Maybe they involve others, maybe it was just alone; but maybe, those are the stories that matter the most.