I haven't played all of the single player FFs, but out of the ones I have played, I would say XIV has quite a few parallels with VI:
-The main enemy of both games is an empire that makes heavy use of Magitek machinery
-Both games feature classic FF spells and abilities, yet their classes are not iconic Jobs (FFXIV technically has Job classes, but lorewise they are extremely rare and are mostly just an extension of their base class)
-Both games have a "Cataclysm" that happens at some point in the story
-Both games have a "War of the Magi" that happened in the past. As a result, Job classes in XIV are virtually extinct and thought to be the stuff of legend, much like magic users in VI
-The Returners and Scions are both organizations of several people that share a common goal and are the main protagonists
-Both games are much darker in setting than is normal for the FF series
-And finally, I've always felt the Player Character has a number of things in common with Terra Branford (especially if you play as a Summoner): For instance the Echo has some similarities with Terra's Esper half, and both characters are even used as a "trump card" against the Empire because of their unique powers
Edited, Aug 7th 2014 3:25pm by JFrombaugh