I'll take a stab at it.
Forgotten Realms:
Several things separate Toril from other campaign worlds or fantasy settings and make it rather unique. One is its sheer size. Even the main continent presented in the FRCS is just mind-bogglingly huge and near to every inch of it crawling with adventur-some goodness. That size lends it two other distinctions. One is an large and very diverse group of threat organizations. The Zhentarim, Thayan wizards, the cult that creates dracoliches (the name escapes me at the moment), the drow, and Mulhorandi are all very different groups with different goals and methods that may plague the PCs at any given time while still leaving plenty of room for all your usual fantasy threat organizations. The second is a list of dieties large enough to fit in such a world, nearly all of whom take a much more active role in the game world than in any of the other settings you mentioned. Finally, FR is such a very developed world. If anything, it's so developed at this point that getting a really firm grip on it as a DM requires a lot of reading (woe be to you if you have an FR novel fan amongst your players). The FRCS has stat-blocks and background information for a ton of very powerful characters that have appeared in one FR novel series or another. This is not a game where the PCs are the world's saviors by default. They have a lot of competition to claw past to claim that title.
Eberron:
In many ways Eberron is the opposite of Toril. There are really only about a dozen countries/regions to host the game. The gods take no active hand in the world if they exist at all. The threat groups, while no less dangerous, are much more subtle in their methods. Magic plays as large (or larger) a role as in FR, but its rarely the PCs' brand of magic. It's simple magic that's almost technology that makes life easier for normal people. The sort of magic the PCs need is actually much harder to come by. Eberron is also notable for being a more morally ambivalent game than FR. The Church of the Silver Flame, nominally a LG organization, went on a crusade that pretty much exterminated all lycanthropes, including the good-aligned were-bears. The country that sued for peace and was most instrumental in ending a century-long war is ruled by a vampire. This sort of fuzzier morality can lead to ethical quandry scenarios that are much harder to achieve in the more black and white worlds of Toril or Middle Earth. Finally, the PCs are heroes (or villans) in Eberron simply by virtue of the fact that PC classed individuals are so rare.
Middle Earth:
Tolkein's world is a much more pastoral setting than any of the others you mentioned. You have a land (Mordor) that is effectively dedicated to evil and the rest of the world is generally a pretty good place. Dangerous perhaps, but not really evil. One could argue that there's nothing in Middle Earth that really constitues arcane magic as we know it in D&D. Gandalf calls himself a wizard, but much of the magic we see him along with his personality is more akin to divine magic. Indeed, all the mithrandir would probably be described (in D&D terms) as good-aligned outsiders from the upper planes. Other than the wizards, magic is simultaneously much more pervasive and much more subtle and usually weaker than in most fantasy settings. Of all the worlds you brought up so far, none is as morally black and white as Middle Earth. Sauron and those in league with him are evil. Everyone else is good, or neutral at worst. Even Denethor can claim madness as an excuse for his behavior.
Greyhawk:
Despite it being -the- default campaign setting, I really know very little about Greyhawk, and will kindly let someone more experienced with the setting address it.
On the whole, the differences you mentioned are mechanical, and you're right. They're fairly cosmetic differences. However, differences in tone, mood, and thematics vary widely from one world to another as does the level of detail (and thus preparation necessary by the DM). Could you move Warforged into FR? Sure, it's just a race. You'd have to come up with some explanation about where they came from, but you're a DM. Being creative is what you get paid to do. Could you move the moral ambiguity of Eberron into Middle Earth? I suppose, but it would probably strain the seams a bit. Could you move FR's very active (I'd say meddling) dieties into Eberron? Not without a lot of work and re-envisioning of the campaign world as a whole.
