You know, one thing that is hard to convey in an MMO is "world-changing."
Some of you have listed tangible things that make "Final Fantasy" what it is (such as crystals, chocobos) but I think that's small picture stuff. The scope is too narrowed.
Personally, I think Final Fantasy has always been great due to a few things. A sense of exploration is one of them. This is XIII's biggest weakness and I think most critics agree the best part of XIII is exploring Pulse. XI did a good job of this as you were barred from areas in several ways - either plot related or due to monsters that would tear you a new one just for trespassing into their territory.
I was Rank 30 when I got to both Camps in Mor Dhona. XIV has good exploration but it's over too quickly (and the fact that once you've seen a zone, you seen about 80% of what that zone has to offer). The ride is simply over too quickly in XIV (so far). It should have been different, more diverse, and/or barred access to pieces. There are plenty of unimplemented things, but that's not what I mean. I mean - something you could work toward to explore a new chunk of the world. That isn't there once you have seen Mor Dhona.
I also think (as I mentioned) player driven meaningful changes to the world aren't happening. We aren't rescuing any Princess Sara(s) or rekindling the light in any crystals. We don't even know who our enemy is. We spend the entire story chasing shadows and pulling at straws. We learn practically nothing. We influence nothing. The finale is basically an epic let down (epic to watch; let down cause you are humbled and have to pick up the pieces). I feel XIV has a strong story and I like where it's going. I expect the next chapter to have some meaningful things but so far it's missing. I think XIV will have a good "Final Fantasy" story once they stop not-telling-us everything and start understanding the Echo, Primals, Garlean objectives, etc.
I think I've said enough for now. I think these are two distinct "grand picture" differences between XIII, XIV and the rest of the series.