The first 5 star in the JRPG genre
Final Fantasy 4 was the first in the series to combine a great gameplay with a great story, the first 3 lacked one of the two, you can call the first of the great ones.
The character design and personality was revolutionary in a RPG at the time. The character driven plot making a dramatic narrative, going for things like search for morality, makes alot of fans of the series consider the 4th game to be the best in the series, and I respect that, it's hard to speak against it.
This game also introduced the ATB (Active Time Battle) system, something that came from Formula 1 racing, according to the designers. And they made it work, giving consistency and value to characters that have speed.
As for the soundtrack, for the first time FF really showcased the enduring themes that later would become synonyms with the series. With songs starting to be adapted and performed by orchestras around the world, specially the song Theme of Love. It's trully one of the best soundtracks ever made, specially for it's time.
What some people didn't like as much, was a more linear story, something that I don't mind at all. Also, the convoluted ending, with some things being exaggerated, what is a normal thing in JRPG's, was a negative point of the game. But still is one of the best RPG's of all time.