I originally posted with a general favor towards Eberron, and I stand by it, but to be fair I thought I'd mention some things I don't particularly like about Eberron and some things I do like about FR. Keep in mind that I have run a good deal of FR but have only prepared the beginnings of a campaign for Eberron.
Eberron Cons:
1) The new races. I really like Changeling, but am fairly ambivalent towards Kalashtar since I don't have the Expanded Psi HB yet. As for Shifter and Warforged, I'm not a huge fan. I really liked Monte Cook's AU, though not Diamond Throne, but I just can't get into fuzzy races. I think it may have something to do with the fact that they're generally represented as being crude dolts, and Eberron's shifters are no exception. I don't mind the race, per se, but I don't like it as a PC option. Warforged strikes me as weak mechanically and kind of boring (mileage obviously varies since many are apparently quite excited by them.) Again, I don't mind the race, but it strikes me as a poor choice for a PC.
2) Religion. I like the very large and well fleshed-out pantheons in FR. It makes sense to me because religion and faith play such a large role in the campaign setting. Eberron's religion seems like it was intended to be less intrusive. However, their pantheons (while small in comparison to FR) seems bulky for the intent. Nine gods in one pantheon, six in another, the silver flame, the blood of Vol, 5 specified druidic orders with textual indications that there are more, etc. That's quite a bit of religion for a fairly sparse population where dieties are gods not GODS.
3) Dragons. I really like the concept of a large group of philosopher/seer dragons who engage themselves with concerns far beyond the lesser races' ken. However, I'm mildly irked that the text suggests that nearly all the dragons live on this particular island/continent that's basically inaccessible to the PCs. It sort of takes the Dragons out of Dungeons & Dragons. I realize I can ignore this as the DM, but we're talking about actual facets of the setting.
Forgotten Realms (Pros)
1) Highly diverse terrain. I, like others, don't like the glacier parked next to the desert, but I'm glad there is a glacial plain and a serious desert area (a few actually). It makes it easy for me to escape the woodsy feel that infuses so much of D&D.
2) NPC Organizations. These abound in FR. Admittedly, there are many present in Eberron as well, but simply due to the longer life of FR, they are more fully developed and you're probably more familiar with them if you've been running FR for quite a long time. They make planning intrigue and adventures easy by supplying you with an extensive list of potential antagonists and supporters for your PCs.
3) High Magic. Magic is prevelant in both settings, but in Eberron it's more mundane. That strikes my mood at the moment, but I do like the intense, in your face feel to magic in FR. I disagree with those who say FR is tolkienesque. Magic in LotR is a fairly subtle effect. In FR, it's huge and everywhere. Dragons lair all over the place, the gods take an active hand in the events of the mortal world, spell casters are common, pockets of wild magic dot the landscape, etc. It's obvious why there are so many adventurers in the Realms. There's so much for them to do.
Anyway, I hope these help you get a feel for Eberron. Henry's right in that you can probably find better info on Wizard's boards. I know there's a primer stickied near the top of the Eberron board there that gives a broad overview.
Z