If we call the world Ringabel is from "world 0" and the starting world "world 1" these are the differences I noticed so far doing the sidequests:
Chapter 5 is world 2, chapter 6 is world 3, chapter 7 is world 4, chapter 8 is world 5
in world 2+ Holly isn't a sadist
In world 2 it looks like Edea was killed by Crowe before attacking the city
In world 3 it looks like Edea was killed by someone before meeting up with Crowe at all
In world 3 Eternia is trying to save Agnes, not kill her
In world 3 Til survives instead of Tiz
In world 2 they got Olivia before you meet her, in world 3 there's no city of the crystalists
In world 2+ if you defeat the swordmaster before the pirate he mentions it
In world 2+ if you defeat the time mage before the rest of the merchant company the fencer doesn't help in those boss fights since he's already dead
In world 3 when you beat the Thief it's revealed he faked his death and has orphans.
In world 3 when you beat Holly+barras it's revealled she used ReRaise and then they quite literally do the deed right there in the cave.
In world 3 Til fought off the Sky Knights by himself which is pretty damn impressive
Those two may be a bit incorrect. From World 2 Crowe's rambling ("You're back!? Why do you hate me so much?"), it sounds like that world's Edea single-handedly kicked his ass and left with the others for parts unknown. In World 2 Olivia's case, the Matriarch says that she died in a random earthquake a few months before while in hiding. Anyways, a few more differences:
- World 2: Einheria is more sympathetic to Edea's betrayal, since she notices the change in her sisters due to their handling of the toxins, but she can't bring herself to betray the Templar.
- World 2+: Victor adds more details to his story about Victoria. In World 2, he reveals that she was once a part of the Orthodoxy, who abandoned her due to her illness. Florem's Matriarch also mentions a former Vestaling who lost her title because she had a serious illness. In World 3, Victor reveals that his father Vincent was actually his assistant. Against Victor's orders, Vincent chose to experiment with alchemical life, "curing" Victoria of her disease, with the side-effects of extreme mental and physical instability. Eventually, Victoria killed Vincent, and Victor decided to devote himself to curing her.
- World 2+: The Matriarch mentions finding a little girl passed out in Florem Gardens, one of the twins that died in World 1 (Presumably, in world 2, they were attacked by monsters on the way and never made it). She also says that the girl has the potential to be a Vestaling.
- Worlds 2-3: In World 2, Qada's subordinate praises him for taking in wounded from both sides. After the subordinate leaves, the party enters and wonder if Qada has really changed. Of course, Qada just admits that he only uses them as test subjects. In World 3, Qada flat-out tells this to his subordinate. After the battle, when the party leaves Starkfort, it's revealed that Qada survived; he begins boasting that he'll create more poisons, kill tons of people, and rule the Duchy. Kamiizumi then speaks from the shadows, chastising Qada's unrepentant attitude. Kamiizumi suddenly appears and kills Qada right there in one swipe.
- World 3: Starting in World 2 I believe, you can start an unofficial side-quest of sorts where you learn more about the history between Braev Lee, Kamiizumi, and Mahzer Lee. There are three people to talk to: An old man in the main Eternian town (Learn about the two heroes of Eternia), an old man in the grave marker village (Learn more about their duel), and then Mahzer to learn the truth. When you go to fight Kamiizumi after learning their history from Mahzer, the pre-fight scene changes. Edea thanks Kamiizumi for his role in bringing Braev and Mahzer together, and for training her. Instead of a battle to the death, Kamiizumi decides to have a friendly sparring session with his student and her friends.
- Worlds 2+: In all of the alternate worlds' Grandships, the Drunken Pig Tavern is replaced by the Sea Slug Tavern. The first time you enter it, a scene plays where the Proprietress from your ship is chews out the Sea Slug's owner over the new tavern's all-around poor quality.
- In some worlds, in the White Magic Tower, a nurse standing next to a little girl talks about the clinic receiving sizable donations from an anonymous donor for a the girl's illness. In other worlds, she says that she doesn't know what it's for, so she puts the money towards treatment for all of the kids in the clinic. Of course, we find out in Chapter 7 that the mystery donor is Khint, and that the girl is probably his daughter. I guess in some worlds, his daughter dies and in others, his daughter lives.