The City of David and the Tower of David are 2 distinct sites in Jerusalem.
The iconic Tower of David wasn't actually built by David: The name comes from the Byzantine Christians who for a long time believed the site to be the palace of King David (the second king of the Israelites). The Tower of David sits in a big fortification near the Jaffa gate that was used by different conquerors over time. The entrance fee included a brief, entertaining, cartoon history of Jerusalem, and a wonderful view of the area.
The City of David is where David built his capital after conquering Jerusalem. Over time, it disappeared from view and is now an active archeological site. There are a lot of ruins in the area, but one of the most interesting area is Hezekiah's Tunnel.
According to the Bible, King Hezekiah prepared Jerusalem for an impending siege by the Assyrians, by "blocking the source of the waters of the upper Gihon, and leading them straight down on the west to the City of David". Now you can walk through the 500m+ water tunnel.... in pitch darkness. At some points, the water goes all the way to your hips (Jen's hips)... :)! It was a fun experience!