When I heard about this game, I did some poking around the interwebs to find out more about it and this is what I've gathered. This is in case you didn't already know, since the info is not hard to find.
As far as the name Bioshock: Infinite. The game takes place in 1912, so calling it Bioshock 3 would be confusing since it is most certainly not a sequel. While there is no ADAM or plasmids, you have abilities called Vigors and Nostrums, the former being the equivalent to plasmids and the latter being like the gene tonics. However, they are not related to ADAM. The reason the Bioshock name still sticks, is because if you explained to someone that the game combat mechanic revolved around finding substances (and drinking them this time) that alters your body to give you new abilities, that person would most likely respond "Oh, you mean like in Bioshock."
While Rapture was created in secret by Andrew Ryan, Columbia isn't a secret and was built overtly by the United States for the 1893 World's Fair. What is secret is that the city was actually built to be a giant floating steampunk Death Star, which I think is really fucking cool.
There are no Big Daddies in Infinite. The figurine in the aquarium was most likely simply meant to be an homage to the original games. The thing kicking your ass in the trailer is an enemy called a Handyman.
The game is even less linear than the previous ones. You can actually return to places you've already been via the sky lines. Also Ken Levine has *hinted* that the decisions you make not only affect whether or not people attack you, but also the the way that the city reconfigures itself. If that is true; awesome!
No more silent protagonist. Yay! I don't know about you, but I'm sick of running around as a fucking mute. This is probably done because you have to communicate with Elizabeth quite a bit. I'm also looking forward to DNF, because Duke never shuts up with the one-liners.
The reason the game won't be released until 2012 is because Irrational had to rewrite a sizable chunk of the game engine to accommodate Columbia's floating dynamics.
Well, I'll shut up now but if you haven't watched the 10 minute early gameplay demo, do so. It explains a lot. Levine also *claims* that very little of this is scripted.
Oh wait, one more thing. The original Bioshock will apparently be made into a movie slated for release in 2013. I know that most game to film adaptations suck, but please Hollywood, don't fuck this one up. Think of the children.