Hi Cutie,
To add to what Galliano and Stouter mentioned, a few things different from FATE Parties:
1. FATEs basically allow anyone to just join in. As a result of this, people don't really have to form parties (they can solo or just duo w/ their Chocobo). There's a sense of "isolation" or not needing to ever party w/ anyone to clear the FATEs (but of course you can if you want to).
2. FATEs are randomly spawned in the zone. While it keeps it interesting (to keep you looking at the map), it also means it's basically a wait-and-see game: See a FATE pop? Mad dash to get to it in time.
When FATEs were really popular early in 2.0, you probably remember seeing mad hordes of players just dashing from 1 FATE to another, slaughtering everything (face roll) and then waiting for the next one to pop. You never really settle in anywhere: Just be ready to run to the next FATE. (It also can be frustrating if you're in a large zone (like Coerthas), doing a FATE in the upper NE area only to see the next FATE spawn in Behemoth's Dominion, LOL. (Groan.))
3. Related to the above is that FATEs are timed. You have to run over and get to a FATE in time, or else it ends, and you have to wait. There's also downtime, when sometimes no FATE spawns for a while, or it's a lower level one in a zone that covers multiple level ranges (I know level sync helps, but it's still less EXP).
XI's Exp Parties were essentially finding a "camp" (area), where there were monsters that were appropriate level to yours and you'd be able to get great Exp off of them. The respawn rate was usually pretty high, so you'd be able to replenish and repeat.
The world mobs in XI were also tougher than current world mobs at your level, so it required a traditional party (Tank, Healer and DPS). I'd say it's kinda like fighting "Go! Go! Gorgimera" FATE (that Chimera clone) in Northern Thanalan (in terms of difficulty), but with a bit less HP. In other words, it wasn't just anyone can face-roll / kill Exp Camp mobs solo (death was a real possibility). You had to have a party to focus.
With "camping," it could get boring, sure, but it also meant more time in the social aspect (being able to chat with the party members), but also XI had a Party Combo System called "Skillchains" where you can coordinate and link certain Weaponskills sequentially, to cause various Elemental Bonus Damage, and then Mages could tack on a well-timed Magic spell to get a Magic Burst.
I can only speak for my server, but Skillchains added a lot of fun and coordination, and kept the interest level up for Exp Parties (you could mix and match various Skillchains / Magic Bursts, etc.). It's more player agency / freedom to engage vs. just the simple individual stuff we do now.
Also, Exp Parties were cool to utilize most areas in the world: FATEs are spread out, but they are always in the same circular spot in fixed parts of the map. With Exp Parties, if there are mobs that are viable all over the map, then any of those areas could be viable to camp in. And it's cool just being out in a certain area (feel like you're exploring a zone).
The social aspect: Perhaps it's a combination of the speed of leveling up now (fast), and anyone being able to Duty Finder / solo queue up for whatever they want to, but with that, there's no real need to get to know many players on your own server beyond your FC / LS. Compared with Exp parties because of the need to get Exp and hunker down in a steady camp spot, you got to start knowing more players (on your own server) and get to build friendships.
Lastly, to those who mention you can just do that now via Party Finder: It's true, but the rewards are awful. The Exp you get from Open World monsters are paltry compared to Dungeon leveling and FATEs, but it was done on purpose (since SE wanted players to experience those aspects over traditional Exp Parties).
Also, Party Finder Dungeon Parties are different because they pull from Cross-Server, and it's all about efficiency (get in and get out). You probably will never see them again, and thus have no reason or feeling to want to socialize.
Going forward, I think it's perfectly reasonable to have all 3 (FATE, Dungeon DF, Open World Exp Parties). I agree that I wouldn't want to see the return of XI's Exp Parties *Only* as the means to level up.
But as *another option* / viable way to level up? Sure, why not? It's all about choices, and SE took the time to make these beautiful zones and fill it with mobs. But currently outside of FATE spawned Mobs, these zones are just "window dressing": Worthless mobs that give you a tiny bit of Exp that make them more of a hindrance than anything.