Job levels vary from class to class. Things like the White/Black mage go up to Level 7, which allows you to cast all six levels of magic and the final level grants a sub ability that grants an MP% Bonus. Some classes like Red Mage though will grant unique skills you can't access. The first three levels grant the access to Lv. 1-3 Magic, but the fourth and final level of the class grants the doublecast ability which lets you cast magic twice per turn. So it is important to keep leveling some jobs because their best skills are at the highest job levels. Likewise, Mastering a job allows the Freelancer and Mime classes access to certain innate skills from the class. For instance, mastering Knight gives both classes innate Cover just like the class, so you wouldn't need to equip the skill on either of them. Same deal with Ninja's Dual Wield skill.

The second reason is because the levels boost your stats when you have a skill from a class equipped. Equ. Weapon skills will often grant the character the same Strength stat as the class, allowing for a squishy black mage to hit a monster with a sword and the strength of a Knight. The higher magic levels you get from the spell casters not only grant access to their magic but also raises their effectiveness. A Knight casting Cure with a Lv. 1 White vs. one casting the same spell with Lv. 6 will show a big difference in power. Finally, Mastering a job grants their highest stats to both the Freelancer and Mime classes. Granted the stats in question will change to a different class if their stats are higher in a particular category. So for example: If you were to master Dragoon, the Freelancer and Mime would get a +18 bonus to their strength stat since it's the highest stat the class has. Now if you were to master Knight right afterwards, Mime and Freelancer would have their strength increased to +23 instead because Knight has a higher base strength than Dragoon.

If you were only playing to get the best stat boosts for Freelancer and Mime, then all you would need to do is master Monk, Thief, and Summoner in order to get the best boosts to Strength, Magic, Stamina and Speed. Unless of course you're playing the Advance/iOS/Steam version of which you need to master Oracle instead of Summoner for the best Magic boost.

If you need an overview of the classes, I've written one up for my Let's Play. I've covered all of the jobs except Freelancer, Mime, and the Advance classes: Gladiator, Oracle, Cannoneer, and Necromancer.