Allow me to provide some of my own insight, as well as some points of my own.
I am currently in the process of making my own video game. It's bizarre and amazing to do, and it's a steep learning curve, but I'm making it happen, but the process of game design is not so crystal clear. Something I think is amazing another might find to be terrible, and something I hate another could say makes it a better game.
For example, do you know what the Legend of Zelda: A Link to the Past team was MOST excited about when they made the game? The graphics? The level design? The story? For Shigeru Miyamoto, it was "the bottle system". Yep. He loved the bottle system most, feeling it allowed players an extra layer of strategy and depth and that people would remember the game for the bottle system. And the composer, Koji Kondo, was most proud of the Cuccoo noises. Yeah, chicken noises over the epic score. Interesting.
Similarly, Silent Hill for the Playstation was a game of compromise. The system couldn't handle draw-distance or smooth framerates, but they wanted a big, open city to explore. Solution? Fog and darkness, giving you a marginal draw-distance and hiding the rest. The results were phenomenal, as radio noise alerted you to monsters, and audio clues and player imagination took over, wondering what was hidden behind the fog and darkness, where it was coming from, how many, how fast, and whether to run or fight before anything ever came into view. It heightened the horror and made it a better game, and it turned something that other games were criticized for into a positive.
But let me answer the questions you proposed.
What makes these games so powerful and amazing?
Different things to different people. Though I would say the most important element for a "legendary" game is "TIMING". A mediocre-to-good game can become a trend-setter if it debuts at the right time, neither too soon nor too late.
Final Fantasy 7 is often called the best RPG ever. Purists know that's not true. Final Fantasy 4 and 6 are routinely acknowledged as better games, alongside Chrono Trigger, or even Phantasy Star II (a main character dies in that game long before Aeris bit the dust), but FF7 was the one that came out at the right time and became the most popular RPG of its time, and the trend-setting that made every game after it (from Legend of the Dragoon to Kingdom Hearts to FFXIII) follow its style and presentation.
Grand Theft Auto III was the 3rd game in the series, but the free-roam, 3D sandbox game was originally created on the N64 by the same team. It was called "Body Harvest" but it didn't catch on. A few years later, they did GTA3 in the same vein and it became a system seller.
Lastly, the Call of Duty franchise is currently the best selling franchise in the world (sigh...), but how many other games predated it? People called EA's "Medal of Honor" a rip-off when it came out... but the CoD team got their start on the original Medal of Honor series, as well as other FPS, ranging from Quake to Goldeneye to Doom to even Wolfenstein. Call of Duty did almost NOTHING new at all... but for some reason it was there at the right time and found the right audience.
What makes them stand out from all other games?
Honestly, the most legendary games on the market aren't very new at all. They innovate little and borrow heavily from other games, as well as their own prequels.
But two things, I feel, make any game stand out: Presentation and Balance.
"Presentation" is how the game is presented (obviously), including how the story is told, how the levels are designed, and even the art direction the game utilizes. A game with good presentation can overcome technical limitations ("Ocarina of Time" was NOT the best looking game at the time) and minor gameplay flaws. Like a tightly edited movie, the game just flows well. For example, the original version of "The Godfather" movie was a failure; critics and test audiences hated it. But they hired a new editor who cleaned up the film, re-edited it, and it became a cultural touchstone of brilliant cinema, all because of its new presentation. The story and acting and everything was the same... but the manner it was shown changed it all.
That leads me to "Balance". Few games have "new" ideas. Ocarina of Time borrowed from prior games, as did Silent Hill 2, Okami, Halo, etc. But what makes a "Zelda"-esque game like Okami better than Darksiders, or a FPS like Halo better than Perfect Dark Zero? Balance. It takes all the elements for success, like a patchwork quilt, and stitches them together in new and interesting ways, but some patterns make more sense than others. Sometimes a game just flows better in one game using the same template than another.
Even between Zelda games, this is true. Why is Ocarina of Time considered better than Twilight Princess? Both of them have nearly the same layout, same methods of progression and even a similar art style. Twilight Princess even has MORE plot and BETTER graphics. So why isn't it as beloved? Balance. Ocarina of Time struck that balance far better, with a world that was big (but not too big), puzzles that were hard (but not too hard), battles that were challenging (but not too challenging), story that was important (but not too intrusive), mini-quests to do (but not too much to derail the quest), and a lot more. Twilight Princess, and Wind Waker too, both got too big, yet too barren. It stretched the fun out between large chunks of progression. Hyrule Field (or Ocean) is bigger, but there's less things to do in it, less secrets per inch of terrain, less people to run into. Darksiders suffered a similar fate; the balance of the game was not as tight and pitch-perfect.
What impacts did they make on the gaming industry?
I actually believe "none". Surprised?
I've learned that the very best games are rarely imitated. The most POPULAR games are always imitated. How does that makes sense, you ask?
Let me ask: how many Ocarina of Time-esque games exist. The numbers are very small indeed. Perhaps, maybe, Darksiders. How many Mario Galaxy-esque games are there? How many Metroid Prime (or Super Metroid)-esque games? There are always a few (like Shadow Complex for Metroid), but they're very rare.
How many Call of Duty-esque games are there? Dozens upon dozens. The "quality" of Call of Duty is debatable, but its popularity is sure-fire. It's the same reason Guitar Hero had so many spin-offs and rip-offs, regardless of its quality.
But there-in lies the difference; it is easy to make a Call of Duty or Guitar Hero or Wii Sports game, taking the gameplay and the aesthetic and skinning it over any FPS or rhythm game. It takes considerably more time, thought, skill, love, and passion to make a game of the caliber of Zelda, Mario, Halo, or Shadow of the Colossus.
And because it takes so much more time and effort, people don't make these games very often. Which makes these games more rare, and hence, more special by their limited existence.
The industry wants a cheap buck, as much money as fast as they can make it, but the true artisans of the industry, the ones with real talent that make the best games, create games that few others would want to spend the time to make. Sure, they'll borrow bits and pieces (Z-targeting, Auto-aim, Bioshock's audio messages, Halo's two-stick control scheme and regenerating health), but they won't touch the "guts" of the good games, the creativity, the passion, or the innovation.
So the impact these games have is... minimal. It's the difference between copying the work of a master artist and understanding how he made it and then making your own.
What sets them apart from their successors and predecessors?
Usually, nothing more than technology and time.
The last Zelda game, Twilight Princess, follows the same template it has since 1986's NES game. Mario Galaxy is still very much the same basic control ideology as the original Donkey Kong arcade game (jump jump jump). Final Fantasy 7 is just another RPG with turn-based battles that does what all its predecessors did (including Dragon Quest, which predates the FF franchise).
But, as I pointed to in "presentation" and "balance" above, it is presented and executed differently. Polished and perfected. And, ultimately, released at the right time at the right audience.
Many games that are failures would have been brilliant successes in another time (Duke Nukem Forever, Shantae, Beyond Good & Evil, Enslaved), while many successes would have failed if they were released now (Goldeneye, Resident Evil 1, GTA3).
A good game series, though, will take the template of the original and improve upon it. Flesh it out. Polish it. Mario Galaxy is the perfected form of Mario 64, which is the perfected form of Super Mario World, which is the perfect form of Mario Bros. 3, which is the perfected form of Mario Bros. which is the perfected form of Donkey Kong. Legend of Zelda begot A Link to the Past begot Ocarina of Time. System Shock begot Bioshock. Body Harvest begot Grand Theft Auto 3. Dragon Quest begot Final Fantasy 7. Alone in the Dark begot Resident Evil and Silent Hill. Ultima Online begot Everquest begot World of Warcraft.
But the reason not every sequel is better is because sometimes a game creator forgets what matters or what was beloved about the game series.
Several recent games have made headlines for "dropping the ball". "Metroid Other M" is almost nothing like the beloved Metroid games it comes from; no exploration, no real puzzles, no self-discovered, limited item hunting, no isolation, a heavy emphasis on story, cutscenes, and combat, and it sacrifices gameplay for narrative, the total opposite of every Metroid game before it.
"The 3rd Birthday" is the third game in the cult favorite "Parasite Eve" franchise, yet it carries over zero aspects of the original games. It doesn't play like the originals, it doesn't sound like the originals, it doesn't feel like the originals. Once strong female heroine Aya Brea is now a weak and sexualized dolt, and even once-likable friends from prior games turn into perverts and even pedophiles for no discernible reason. It is a game that actually is LESS fun if you've played prior games and know the history of these characters, and how they are systematically betrayed and ruined by the creators.
But look no further than the "Silent Hill" and "Resident Evil" franchises to see two games that once thrived in their genre but that have become tempered and redundant. Silent Hill's creative team was disbanded and the franchise given to Western studios that mistook gore and gunplay for horror and atmosphere. They felt making aggressive enemies and trying to replicate SH2's successes would lead to similar results, but they've found out the hard way that a Pyramid Head cameo and more blood doesn't make a scarier or better game. On the opposite end, Resident Evil ditched it's horror roots entirely when RE5 came around, providing zero scares as you mowed down dimwitted body fodder with a gatling gun and rocket launchers alongside a dimwitted partner the whole time. Fans hated it, and I doubt anybody will be trumpeting RE5 as a "legendary" game alongside RE1, RE2, and RE4.
But as every franchise wanes and waxes, new ones arise that take up the spirit of the flailing franchises and does something new with the old ideas. Batman: Arkham Asylum uses the Metroid formula of exploration, self-discovery, and self-improvement, meshes it with Bioshock's audio logs and Eternal Darkness's sanity effects, then even adds a Guitar Hero-esque rhythm game to the combat and the results were amazing. Dead Space took the best parts of Resident Evil, crossbred them with elements of "The Thing" and "Aliens" in space, and then created a HUD-less interface that kept players immersed. Mass Effect 2 builds from its RPG pedigree from the likes of KOTOR and Jade Empire, polishes it with fast-paced Gears of War-style shootouts, and even drops in an addicting romantic dating simulator game between missions (Miranda or Tali?), all the while taking the best bits of Star Wars, Blade Runner, Star Trek, Battlestar Galactica, and more to populate its sci-fi world.
There are no new ideas, only new mutations of these ideas. But every franchise has its "peak", the moment when it found the perfect "balance" and "presentation" of its award-winning formula. Some games are lucky enough to have multiple peaks (A Link to the Past=Ocarina of Time, Super Metroid=Metroid Prime, Super Mario Bros 3=Mario Galaxy, Resident Evil 2=Resident Evil 4), while some are one-off wonders (Okami, Psychonauts, ICO, Enslaved), others linger with residual popularity of serviceable quality (Madden, Call of Duty, Guitar Hero), and others eventually peter out and become tired shells of their former glory (Sonic games, Tony Hawk games).
Concluding, what makes a "legendary" game is a combination of things, some based on talent, some based on timing, some based on luck. And even then, time may paint a far kinder picture of some games than others (people were much kinder to Perfect Dark Zero and Black & White when they came out, and much tougher on Team Fortress 2 and Deadly Premonition, but the latter games have developed passionate fanbases).
For my own game, it's going to be a combination of ideas that I know work well, gambles that I hope pay off, and hope that the timing is right and the market is ripe for the type of game, story, and genre I wish to explore, and even with a solid grasp of this, one never knows how the game ultimately will be viewed... not until time passes and reflection can be spent on its merits and detriments in equal regard.
