They take place about 50 to 60 years apart.
BioShock and Infinite of course, but how different are these places really?
Columbia is a paradise and Rapture is a hell.
Think of this theory: Booker DeWitt ends up in a room tainted by Elizabeth's tears, she brought something back from the future and her Cinderella story happens. Booker is attacked by Songbird, thrown out of Rapture and is saved by a Tear. In Songbird's quest to get Elizabeth back, he tears apart Columbia and chases Elizabeth back to New York where it is finally destroyed. The U.S Government then takes Columbia out of the sky and manages to cover everything about it up, its gone from U.S History. That done, evolution happens and vigors evolve into a new substance from which a very powerful substance can be made. Then one man finds out about Columbia and decides to build Rapture: Andrew Ryan upgrades Columbia into something more modern, Rapture. He manages to turn what the vigors have evolved into a substance called plasmids.
Infinite finishes and BioShock starts.