That depends. First of all, mastering a job means the character has learned all of that job's abilities and he or she can't advance in it any further or spend any more job points on it, as there are no abilities left to buy with them.
Advancing a character's job levels opens up different job classes... for instance, achieving level 2 in the Squire job unlocks Knight and Archer for that character. The Dark Knight job, new to the PSP port, includes among its unlocking requirements that the character master both Knight and Black Mage.
Another new job, Onion Knight, makes no use of job points and can only be leveled up by mastering other jobs; for every two jobs mastered, excluding Squire and Chemist, the character gains one job level in Onion Knight. Onion Knights are arguably useless until they reach job level 8.
In most cases, mastering a job is not strictly necessary unless you really want the character to have all of that job's abilities available to him or her.