The classes are things just about anyone can learn if they apply themselves - no real restrictions on who can become one, or difficulties in finding a teacher.

Jobs tend to be far rarer, and often restricted one way or another.
White Mages for example have been pretty much illegal to everyone except the Padjals ever since misuse of White and Black magic caused a prior calamity. An exception is made for the Warrior of Light just because they are so super-special.
Bards on the other hand aren't illegal in any way, but there are very few of them around anymore who can teach you. The guy you find to teach you is just about the only qualified bard still around.

That's the difference story-wise anyway.


Jobcrystals work (in-game) to accelerate the learning process so you can learn in a matter on months what should take years.
Mostly they are a game-mechanical thing which shouldn't be thought too hard about.

As for why the newer jobs do not have a base class, that is simply because the current developer team does not like the class->job progression and would like to do away with it.