Are the Final Fantasy games interconnected or independent, story-wise?

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

Exceptions:

• 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.

Short answer: games are all independent story wise (except 2 "sequels") but there are recurring themes and elements. You can play them in any order.
Recurring elements are in the form of names, species, ennemies, items, game concepts... Enough to know in a few seconds you are in a FF game, but not locking the story in a specific world or era.
Story wise you'll generally end up having to go on a quest and assemble a team to fight some kind of great evil or villain in order to save the world. Even though it generally revolves around a positive message of love/tolerance or environmentalism, some also have darker elements or interpretation.

Gameplay-wise they are traditional JRPG up until FF10, then it's a bit more modern/dynamic (while still being way more classical RPG than action RPG).
Character progression system is different in each game (some games are closer than others though).

Long answer : https://en.wikipedia.org/wiki/Recurring_elements_in_the_Final_Fantasy_series