As you already said, the games are usually set in a different world, but some elements are common in all/most those worlds.
- Chocobos (an animal that appears in all (?) version of final fantasy)
- magical domains are similar in many of the games (most specifically black magic)
- some items reincarnate in similar ways (e.g. revival items)
- powerful beings that can even be summoned (many element-based). Sometimes they need to be defeated before "taming" them
- main character usually wields some kind of sword
- FF 4 has a sequel spin-off (2 chapters)
- FF 7 has several sequel/prequel spin-offs
- FF x and ff x-2 are set in the same world (lore-wise). Gameplay wise they are (sadly) entirely different.
- FF 12 has a sequel spin-off
- FF 13 has a sequel spin-off
It is also worth noting that some of the more recent releases of the franchise are MMO games. (FF 11 & FF 14)
Technically the games are usually quite different from each other, while still having a firm common core.
Common core mechanics:
- Party-based, usually 3-4 heroes, larger hero pool than party size
- Heroes unlock as story progresses, sometimes certain heroes are unavailable
- Heroes develop with experience
- Heroes wear certain equipment usually weapon and armor, sometimes rings. Weapons and armor are usually class specific (which in many cases means hero-specific). Weapons usually have no (negative) tradeoff effects, so every weapon is an upgrade
- World is layered in: World for travel and chance encounters, City/Dungeon for Story progression and exploration and Combat which opens up a small fighting close-up when you encounter enemies
- Combat is pseudo-real-time with turn-based elements. Every hero has a certain time that he needs to regain an action.
- each action can be either attacking, casting a spell, using an item or using some special moves. The player can choose which action will be performed.
- maximum damage and health value is either 9.999 or 99.999. In some cases there is a hidden upgrade to expand to the next higher 999... limit.
- Epic bossfights. Very powerful bosses have either several components with their own health bars, resistances (and attacks) or several incarnations or both.
Significant differences (examples):
- ff 7: materia system, spells are bound to spheres, that you need to socket into your gear and that gain experience independent from your heroes. All in 3D.
- ff 10: Sphere board for character progression
- ff 13: only one character can be controlled, the other two are automated. Dynamic changes of "classes" during combat.
There are also some common themes that reoccur quite often:
- big corporations/governments that oppress or endanger their civilization
- hero is usually linked with the inner circle of the bigger evil (family, childhood, destiny)
- unethical experiments/research by evil oppressor
- value of balance over dominance (e.g. between good/evil, life/death, elements, etc.)
- art-style has many influences from asian all the way to european gothic. Not excluding entirely new creations.