Here is what I think is so special about Forgotten Realms (pre 4e):
Among the people I game with, it is the standard D&D setting. Dragonlance, Greyhawk, Eberron, they all are D&D settings, but seem like "other" choices. To me, and my friends, the Realms is just the natural setting for D&D. (Yes, I know it wasn't the first, and Greyhawk references are in the core, I'm just talking about "feel" with this paragraph). When 3e was first announced and I first read of the previews for the Spell Domains mechanic for Clerics and that there would be deities in the PHB, my best friend's first question was: "So, what Domains will Lathander grant?"
I like the "Kitchen Sink" feel, it doesn't feel like one person's work (Yeah, it started with Greenwood, but how it's grown!), so it feels more natural and organic as it's grown and pieced together over the years. Some other settings are pretty monolithic in their style and flavor, but the Realms are all over. You've got your standard pseudo-medieval Tolkienish fantasy with Cormyr, Sembia and the Dalelands. You've got lands patterned on India (Durpar), Ancient Greece (Chessenta), Ancient Egypt (Mulhorrand), Babylon (Unther), Celtic lands (Moonshaes), Korea, China and Japan (the various nations of Kara Tur), Mongolia (Hordelands), North Africa (Calimshan), African Rainforests (Chult), Spain (Amn), the Middle East (Zakhara), piracy-era Caribbean (Sea of Fallen Stars) Central America (Maztica), frozen lands (Vaasa and the High Ice) with a significant list of purely fantastic high-magic lands of several styles (Thay, City of Shade, Halruaa), typical racial homelands the PC's expect (Lantan, Luiren, Cormanthyr & Evermeet, The Great Rift), and a huge city that would rival Imperial Rome (Waterdeep) and you could run a half-dozen campaigns in that city alone without ever really leaving. Pretty much any D&D character concept you could come up with could fit in somewhere on Abeir-Toril, and pretty much any adventure you'd want to run could be done somewhere. Saying "that doesn't really fit into the campaign world" is hard to say (but "that isn't the kind of thing I want in my campaign" is always DM's prerogative).
The Flavor! 2e was especially good with this, with the Volo's Guides, and Aurora's Whole Realms Catalogue. Forgotten Realms books tend to be wonderfully immersive, things you can use to really feel like you're "in" the world and can understand the cultures and places and people. It feels more like a complete world and less like a created backdrop for adventuring.
The Pantheon. Personally, I always liked the Faerunian Pantheon as a set of D&D gods. They had the foibles and politics among them much like the Olympian deities, but had such wide variety and selection for players that there seemed like there was always some deity/faith that fit a character concept. The faiths and religions of Faerun were a lot of the appeal of the Realms to me.
The detail. I loved the detail of lore and setting information. Any town on the map I could point to I could probably pull up a map of, detailed information about, and probably lots of obscure minutiae. I never had the problem of players who tried to setting-lawyer me. All the people I ever played with seemed to understand that no one person will know all the setting, it doesn't matter if this town doesn't have a shrine to this deity like this book said it did, or that this NPC is secretly working for this organization even though they just proved he couldn't be, or that if they go to this obscure island and walk to this specific spot and dig 10 feet down they won't actually find the artifact that is buried there even though some book says it's supposed to be right there. I tried to always run the Realms in a way that was fun, and to try to not completely trample the setting (to maintain the shared-world feeling with the players), but to not let it hinder me either (to maintain my freedom as a DM). The detail is there so as a DM I have a vast well of ideas to draw from, not as shackles for the players to bind me with.
I could probably keep going, but these are the big reasons why I love the Realms.