I'm a Realms fan, I make no secret of that, so I'll tell you what turns me off from Eberron.
A race of sentient golems. Playing a fantasy robot PC just seems a little too wacky, and the entire warforged concept really stretches (breaks actually) what I'll consider themati for a "normal" D&D setting.
A D&D version of Coruscant, with the insanely tall towers we can't engineer today with 21st century technology and a population density that makes those towers more empty than farmland.
The nonsensical demographics, magic is insanely common, but PC's that should only be roughly coming of age are supposed to be world class. Even though magic is insanely common, it's hard as heck to get divine magic cast.
The lack of real gods to the religions and the lack of alignment restrictions on Clerics. I don't want moral ambiguity in my clerics and good faiths with evil congregations in my games, I play D&D to get away from things like religions of peace being perverted into crusades and torture and hypocritcal clergy who pervert holy books and use a good name as an excuse for hatred and intolerance.
"Design by committee" feel. Eberron was made so that everything from the Core Rules could be used in it, and it shows. You can look at it and see the part that was bolted on here so that you could have this here, and here is this, and we had to make a way for suchandsuch to exist, so we came up with this.
No real personalities. I think it was supposed to be a selling point of Eberron that their aren't any major, powerful NPC's, but I see that as unrealistic and frankly a little boring. Eberron seems highly contrived to make PC's the center of the entire world, even if they shouldn't be. They should be the center of the game, but not of the entire campaign world (unless it's an Epic game).
And I'll speak in favor of the Realms, why I like it.
Analogs of real world cultures exist. If a PC wants to play an egyptian themed character, he can be from Mulhorand, or be from Zakhara or Calimshan if you want an arabian character, and so on. It helps PC's and DM's get a quick grip on a kingdom if it's thematically similar to a world they already know.
It feels like a real world. It's got an unparalelled breadth and depth of setting, you can point almost anywhere on the map and have reams of information available. If you don't want to read through it all, you can get a feel for the basics quickly and work with that, but if the PC's decide to go to some random town on the map, you can know the local culture, history, politics and the like very easily. There may well be a powerful wizard or a renowned warrior who lives there, but he won't be at the PC's beck and call, he's got his own battles to fight and he's off adventuring too.
More than one flavor. Eberron is going for a dark fantasy/pulp look and feel, and the Realms are more of a traditional tolkienesque high-fantasy, however Eberron in its theme is pretty one note, while the Realms do have places that are darker, and places of wonder and discovery, and while both Eberron and the Realms have been written by many people over time (based on one persons concepts), the Realms shows more distinct voices in my experience, and it seems like it's not a one-note setting.