Difference between a LS and FC

[Niwashi]

Holy hell lol. So basically don't look for a LS or FC to join until I've hit level 50? That's an awfully long time of me, myself, and I.

No. You'll want to be in one by (or soon after) hitting 15. That's where the group content starts, and FCs or LSs are the best places to find people who will run it together with you the way you want to run it (though the game does have a feature for putting you together with random other players if you don't find people yourself). Take some care in finding an FC that suits your interests and playstyle, but don't put it off too long.

do you basically play the main story by yourself until level 50 when you apparently unlock what seems like the "real game"?

As mentioned, group content starts at 15, but more significantly, no, the "real game" here doesn't start at 50. It definitely starts at 1. If you get well into mid-game and the story still doesn't interest you, then it likely never will.

The game's tutorial is only a single quest, done in a solo instance that's completed before you even enter the game world. Later, there are some significant milestones, like at level 15, when airships are unlocked, the three city-state storylines all merge into a common storyline, and dungeons open up. At 20, you'll join the Grand Company of one of the three city-states of Eorzea, and that will bring in mounts and a new type of currency to earn and spend. Then another milestone occurs at 30, when you unlock the job(s) that advance or specialize the class(es) you played up until then. Level 50 is yet another milestone where more content opens up. It's one more step along the way, but neither the beginning nor the ending of the game.


EDIT:
Oh, and I guess I should answer the main question of the thread as well:
Linkshells are just private chat channels and that's it. Free Companies have a private chat channel, but they also have an inventory chest for sharing items, and members' actions within the game earn the FC points that let it increase in rank, earning other priviledges. They have identifying crests that members can wear. They can have a shared house together. They can spend action points to activate any of a wide range of bonuses for all members (XP boosts, travel discounts, reduced gear wear, crafting bonuses, etc.)

Those things are among the differences that are explicitly set by the game. There is also a strong tendency (though with some exceptions in both directions) for FCs to become more close-knit than linkshells, but either FCs or LSs can be as specialized or as broadly based as its founders and members want it to be.