Two fairly-recent games which truly deserve attention are Riviera: The Promised Land and Yggdra Union, both GBA games published by Atlus and developed by Sting. Abstractly, both games do two main things: take fairly-established genres with a strong (as opposed to trivial) twist, and have enough small (here, almost trivial) twists that combine to create character. The first is a quality necessary for creativity, the second is important for the game to become a classic.
Let me expound on them a bit, starting with the first quality. Riviera is best described as an adventure, of the choosing to move around, selecting branching options type, with the important twist that it has RPG-style battles and character development. Yggdra Union is a fairly standard SRPG, where you move armies around and engage in battle, with the twist that you choose a card each turn and may only fight one battle each turn, which turns the game into a more-chess like tactical battle, where numbers matter a lot less than tactical placement. What does this mean for the games? It means that for Riviera, you are not doing a lot of leveling up (you actually can't, since it is turn-based, as I'll explain later), so the battles don't create tedium - yet you still care about advancement, so things move at a good pace. It means that for Yggdra Union, you can't charge around randomly, you need to stop and plan your moves, something lacking in many SRPGs, which makes the "S" meaningless much of the time.
The above is cool - it is what makes many games good. X-COM was good since it takes the Civilization mold and adds a tactical component, for example. But my focus will be on the second trait - what do I mean small twists that give character? These are very brave decisions on Sting's part to add blatantly unrealistic elements into the game that somehow make sense in the game world. Some of these things we take granted (like playing games in 2D, for one), but after we've taken video game physics and video game logic for granted for a while, things stop becoming fresh. Not Riviera and Yggdra Union. The most representative such feature actually occurs in both games, every item (even weapons) can only be used a finite number of times (in Riviera this is measured by turns, in Yggdra Union battles), which makes some weird sense - but not only that, it adds greatly to gameplay since you have to plan ahead. A lot. These changes daring enough to change the way teh gamer looks at the game, but not so outrageous as to turn them away, are incredibly different strokes from the designer to come up with - they really reflect artistic mastery in the designer. Another example that is common to the two games is that both your characters and the enemies can use "desperation moves" in combat depending on the "rage bar," which is borrowed from fighting games. I know FFVII did this for the players, but have you seen the enemies with it? It adds a new scope to the tactical battles since sometimes the enemies' desperation moves are quite potent, so timing victories to avoid them becomes important.
I'll now give some game-specific examples. In Riviera, you have weird things like being able to "level up" a character's affinitiy with an item (everything from a weapon to milk) which gives the character stat/skill gains (so if you use a weapon 10 times, or a herb 5 times, you gain some skill/stat). My first (and probably many's first) response to this is "wtf? but I guess it sort of makes sense," which is the genius of the situation: after the initial puzzlement, the player forgets this incongruousness later and then worries about things like spreading out the item's (finite) uses so that everyone can get stat gains/skills, especially since different people use different items differently (for example, one item protects the party when used by everyone except when used by one of the girls, which casts a spell). As my literature professor once said: "great writing makes the reader suspend his disbelief long enough for the writing to pull the reader into his world." This is totally true for this game. There is also a hidden romance meter (remember FFVII?).
For Yggdra union, you have these cute features like "unioning" (depending on the gender (...) of your leader, when one army assaults another different units will join in the fight, depending on if they are in the area covered by a certain formation with the leader as the center... this is damn hard to describe) which totally changes this from a SRPG to a thinking game (it is still easy, but finding optimal moves require some combinatorial planning, and they are very rewarding when you find them). Also, there are little things like time of day, towns to visit, etc. The coolest aspect is the cards, since each card immensely affects what happens in that turn (and the enemie's turn following it). They determine the total number of points you may spend on movement (yes, a total number of points), who can use desperations, what the desperations are, and how much damage is done with the cards (which level up with use). My descriptions don't do the game justice - so you have to try to play it to see exactly what I mean - the cards make a strange video game logic that first perplexes the player, but then is ignored by the player, who ends up thinking of strategies coping with the logic. Also, your troops have morale (like health) which are NOT restored between missions - unless you get rid of items, which give different people different morale. Again, this makes no sense, but you soon get used to it.
In one line, I can describe this phenomenon as what makes Magic: the Gathering special (Besides the collecting component). It offers a scenario where the cards and their context don't really make sense, but they make just enough sense for you to start thinking about game logic instead of whether they make sense or not. Appreciators of the game should understand this feeling completely.
Of course, I have not mentioned the beautiful artwork, catchy music, funny Japanese-style dialogue and item descriptions. If you are treating this as a review, go play. If you aren't, then hopefully I've captured one weird aspect that makes certain games unique. I'm sure I can generalize this concept to other classical games if I tried hard enough.