Yes. They both exist within the game's universe, but since Rapture was built much later than Columbia, almost no-one in Columbia knows about Rapture. It is revealed via audio recordings that select few from Rapture looked into the future Rapture and took advantage of some things that were revealed to them in this way (mostly technology).
Players return to Rapture briefly within the game, but it is not important to the general storyline. The game does mention the fact that every dimension has their own version of Rapture, as well as a Ryan and Fontaine.
The two DLC epilogues does in fact lead directly to the events of Bioshock 1.