1. Houses & Alliances
In Game of Thrones (GOT), all the main characters belong to prominent families, known as "Houses". All of these houses wield consider political, militarist and economic power. Some houses are the ruling elite of kingdoms, while others are incredibly wealthy. Together, these different houses form alliances to run empires. This causes considerable allegiance conflicts between loyalty to a house/family and allegiance to the throne. In turn, rivalries are formed between different houses.These type of political house of cards are nearly nonexistent in Dungeons & Dragons styled RPGs. In D&Desque games, the conflicts are usually external (e.g. orcs attacking a village, a rogue wizard trying to open a portal to a chaotic plane of existence, etc). The various kingdoms within most Game Master's worlds reside in the background; serving as a quest giver, or swift arm of justice when the party gets out of hand.
However, the dynamic of allegiances and rivalries can not only serve as an interesting backdrop, it can be the primary driver of the plot. Player-Characters (PCs) starting as honorbound knights can serve a house (or king). This provides players with a sense of duty for their PCs to full-fill (rather than the done-to-death merc hobos looking for work). Additionally, making the players choose where their PC's allegiance truly lies creates a fertile soil for moral dilemmas and strategic planning of the future.
2. Secrets & Betrayals
In most RPG adventures the objectives are pretty clear: clear out the dungeon, defeat the big bad monster, or acquire a priceless artifact. But in GOT the plot (and everyone's goals) are constantly evolving, reshaping and changing. One king takes the throne, then is backstabbed, only to be replaced by another. Secrets are the currency of the powers-that-be; and knowing who to trust them with and how to wield them is the name of the game. This creates a sense of a living world, full of other people; each with their own goals and hopes of achievement. Learning who is on your side and who is secretly plotting against you is a very useful skill.
Adding these elements to your own RPG adventure will bring the PC's charisma score front and center; and in some ways perhaps make it their most important attribute (instead of being considered a "dump" stat for most players).
Betrayals also play an important role throughout the series. Think of them more like "power plays"; as a swift and deadly move of a rook across the board in an attempt to take the throne or place another side in check mate (forcing them to bend the knee). PCs can also engage in this sort of behavior to move up the social ladder, acquire more wealth, uphold a loyalty pledge or seize a throne.
3. Low Fantasy
When I first started watching GOT, I actually thought there was little magic or fantasy in it. But the more I watched, the more I realized that there certainly is magic; but its kept low-key. Mentions of dragons, giants, undead, blood magic and so on add a nice garnish to an otherwise human-centric TV series. No elves, nor dwarves nor halflings.
Instead, the magic that is displayed feels more special. It gets more focus, more details. It feels more down-to-earth and even plausible in some ways. Most D&D style games, in contrast, have huge amounts of magic. Level 1 characters can start out as wizards, and magic is used throughout every demographic of society. It's always felt a little strange, to me, for so many people to know magic; but not see magic dramatically change the human condition of the in-game world.
In Game of Thrones, very, very few people know or understand magic. Mostly sorceresses. The dragons were thought to be extinct (the same with giants and other large mythological beasts). Humans reign supreme, and the focus is squarely on secrets, steel and steadfastness. People have to rely on their wits to keep their mortal bodies from easily being destroyed. No healing potions, no magic missile, no fireballs.
4. Morality
Game of Thrones is a very adult show. There are many depictions of brutal torture, executions, war, treason, rape and so on. It paints a picture of a cold, cruel world of survival. There are many, many people in the show who are evil and revel in sadistic pleasures. But there are also genuinely good, honorable folk who refuse to sacrifice their honor for power.
For me, it's a breath of fresh air to see such a raw, dramatic depiction of human nature. Moral and ethical choices are constantly being made by everyone, and the consequences of their actions soon follow. It all makes for a dramatic show, but also leaves me pondering my own thoughts and feelings about certain moral situations. Would you lead a coup to overtake the throne; if it meant you would be a more benevolent leader than the king's heir? Would you execute your rival if you felt they were truly evil and could not be reliably kept under guard? Would you choose to keep your honor and integrity or lie and aid an evil force to save your family?
Try adding more moral dilemmas to your game, and you'll be offering something far more deeper and rich than just killing vampires, lichs and zombie hordes.
5. History & Nature of the World
Any D&D player will recognize certain, familiar elements within Game of Thrones. One of my favorite is the "Night's Watch", a military unit tasked with guarding a northern, icy wall from invaders. Within this military force is an elite unit, called the "Rangers". In D&D there are also rangers, but the background history and story of them is typically left up to the GMs to decide. What I like a lot about the rangers in GOT, though, is the amount of history added to them. We don't just hear about the rangers, we meet some of them. We see how they train. We learn how they travel beyond the wall--and into the snowy wilderness--for months on end as scouts. They talk with "wildlings" (natives of the north), gather intelligence, and report back to Winterfell. They even have an awesome honor pledge they take to protect the world from invaders. Adding an entire backstory to various orders, is what makes the world feel alive. Details about not only what they are, but why they do it, how, and the conflicts they face.
The same can be said for the world as a whole. Their world operates much the same way as normal earth, but with a few twists here and there. For example, their seasons can span for many many years. Winters can last a generation. These twists add a nice "theme" to the setting, and allows for the viewers to explore how the characters react to such changes. This reminds me of the "Dark Sun" campaigns in D&D, which are similar to a generic fantasy--except it takes place in a post-apocalyptic desert world.
It's this extra level of detail that, to me, makes the whole world feel a "cut above" more generic adventures.