Final Fantasy III is a bit unique compared to other FF games, portwise.
The Famicom version is FFIII in its purest, original form. As the earliest form of the Job System, Final Fantasy III walked so that Final Fantasy V could run.
Final Fantasy III was the third and final of the "early era" of Final Fantasy games and most of what II didn't establish was established here. III is something of an evolution of the first game: The Elemental Crystals return to prominence after their brief supporting stint in II and plot goes back to four people exploring the world to foil the forces of darkness. III's primary mechanic is its job system: Rather than picking a class and sticking with it for the entire game, getting an upgrade partway through, or building up your characters based on their actions, III allows the player to change the jobs of the characters at will, easily change their role in party (although being harder to do so without reason).
And III really likes doing this.
Ultimately, III is a first try at the job system formula. Plotwise it is probably one of the weakest in the series, being less a single overarching plotline and more a string of adventures and escapades until the last fourth of the game. Unlike Final Fantasy II, and for the last time in the mainline series until the MMO entries XI and XIV, III lacks explicit player characters, though they aren't quite the total blank slates that FFI has. Gameplaywise, the job system is fun to toy around with, but the nature of III's gameplay means that you'll inevitably run into a section that requires a specific job to get through. It's not impossible to play around these sections, but it's not easy to ignore either. Still, it lacks the larger issues that Final Fantasy II has and has more meat on it than the original Final Fantasy. It arguably is one of the finer NES RPGs and is certainly one of the beefier ones.
Around the 2000s, Square Enix decided to redo the first three Final Fantasy games for the Wonderswan Color, a portable gaming system. While Final Fantasies I and II made it onto the system, III did not.
The ramifications of this are staggering, if one were to think it out heavily: Without a Wonderswan version, there was no basis for potential PS1 or GBA versions, leaving a sizable hole in the collections on those systems. It was III remaining "lost" for so long that led SE to remake the game, expaning on the lore and adding four new characters to function as the main protagonists. Leaving aside the knock-on ramifications of that, the devs would then go on to remake Final Fantasy IV before going on to make an original throwback game, Final Fantasy: The 4 Heroes of Light. They would then make a sequel to that game, which would become it's own beast named... Bravely Default.
How's that butterfly effect for ya?
As for why it got canned, it's not completely clear. Some say it was technical difficulties, while others say it was because SE lost the source code. The officially given reason is that III was too big to get done, although it doesn't seem to be a space issue as even FFIV did not take up that much of a Wonderswan cartridge. In any case, it didn't happen and, in a twist that is on brand for this game, its history took a different turn...
In 2006 or so, Square-Enix noticed that they hadn't touched FF3 for a solid decade or two. Deciding a remake was in order, they targeted Nintendo's hot new handheld at the time, the Nintendo DS, at the Big N's request. Compared to the remasters that the earlier games got on the PS1 and GBA, the III remake was significantly grander in scope and detail, making changes far and wide. The art of the game was remodeled by Akihiko Yoshida, going for a chibi look with a pop-up book-style world. The music went for a faux-orchestral vibe. And the plot and gameplay were both heavily overhauled.
The storyline's biggest change relates to the main characters: In the original game, the main characters were four non-distinct Onion Knights. Uusually, if all four didn't speak, then whoever was in the lead would. For the remake, the main characters were changed to be given distinct identities, personalities and backgrounds: An impulsive youth named Luneth, his bashful bookworm of a brother named Arc, the caring if short-tempered Refia and a stoic soldier from the nearby castle, Ingus. It was intended to add numerous new scenes to the game, but by the time the final game took shape, it was decided to prioritize scenes that were already there to begin with. Still, there were also a number of scenes, major and minor, that got changed in ways to flesh out the story or to tighten the narrative.
Gameplaywise, there were quite a few differences. On the positive side, the philosophy of the remake was to improve jobs to such an extent that the game can be cleared with most of them. Thus, many jobs got new skills, new weapon/armor compatibility and overall new leases on life. Guest characters also now help out in battle, although only at random. On the negative side, the nature of the game's development (namely the cartridges used) means that only three enemies can be rendered on-screen at a time (discounting a fight in the Amur Sewers). This, combined with the overall perception of FF3's difficulty means that the final game is more of a challenge compared to the original: There's fewer enemies, but those enemies can hit harder. Bosses in particular can attack more than once, taking two or even three turns. It's a harder game, although there are also better tools at your disposal to fight back. The most contentious factor is the Job Adjustment Phase, a replacement for the Famicom's "Capacity Point" system; Upon changing a job, the affected character must go a number of battles with reduced stats. On one hand, there's no possiility of being "locked" out of a job and it actually works as a penalty. On the other, it arguably slows the game down and the fact that there is a penalty to begin with is contentuous.
The Final Fantasy III remake was later ported to PSP, mobile and PC. The PSP version added a few goodies including an art gallery, an autobattle function and the ability to switch between the original Famicom soundtrack and the remake's soundtrack (CD quality compared to the DS version, which was native). The PC and mobile versions were eventually brought to a similar standard, only missing the alternate soundtrack. The biggest difference between the original and the ports is the removal of the wireless component: The DS version had the Mognet system which allowed players to message each other. A number of in-game sidequests were tied to this system, namely the Onion Knight job, the Legendary Smith questline and the Iron Giant superboss. Later ports simply tie progression to the lead character and the progress in the story.
The remake's overall perception these days is very "love-it-or-hate-it". Some like the job changes, the plot changes and the overall package. Others dislike the high difficulty, feel that the new main characters are too flat to care about and overall prefer the original.
In 2021, Square-Enix announced a series of "remasters". Dubbed the "Pixel Remaster" series, this series boasts 2D visuals and an attempt at toeing a fine line between being closer to the original and having quality of life changes. To that end, the series reverts the FF1 Magic system back to its original "charges" system compared to the GBA version's "mana point" system and cuts all of the content added in the re-releases. Final Fantasy 3 is part of this line, meaning that this is the first time that the Famicom version of the game has been re-released.
... Except, not really.
The Pixel Remaster of Final Fantasy III is something more akin to a missing link between either version. The script and level design is based on the Famicom version, with none of the remake's changes. The gameplay, however, has some fundamental changes: Some jobs, such as the Scholar, get skills based on their remake counterparts. Some of these skills have changes: Bard's songs are now connected to Job Level rather than harps, Dark Knight's 'Darkness' command is now 'Bladeblitz' and costs no HP, etc. And a rare few jobs have new abilities, such as Black Belt, which now has the kick command from later FF games such as IV and V. One particular change is the Job Change Penalty: Because there is none. This is a bit bittersweet; On one hand, it is massively convenient and III in particular places emphasis using certain jobs in certian spots, making that more managable. On the other, unlike FFV, the game's encouragement of switching jobs comes less from the skills you can gain and more from the situations it presents to you, which brings the possibility of certain jobs being more situational than they otherwise would be. On paper, anyway.
Graphically, the game varies: Most graphics are based on the Famicom version but with more detail. However, there are certain areas, particularly buildings such as the Dragon Spire or Tozus, that take cues from the remake's interpretations. Then there are a few details, such as the flags of Argus and Saronia that are unique to that version.