My two copper pieces.
Basic Idea: There's no single one (that would span all the PrC's), but several:
They are there to make your campaign unique, to give the player a aim to work towards, and to give them unique powers.
The Difference between Kits and PrC's: Kits were altered classes, while PrC's are new classes (though often based on one or several classes). Kits were there to be taken from the start, PrC's can't be, cause you must meet the prerequisites first. Kits were taken instead of the base class, PrC's are to be multiclassed in.
Reasons for PrCs:
- Racial PrC's fill a special role in the society of that race: There's the elven warrior/poet (bladesinger) or dwarven wild warrior (battlerager).
- Special Societies can have their own PrC, often with the prerequisite to be member or that society: The Assassin, the Guild Mage, the Purple Knight, the Divine Champion.
Both of these will often grant you some prestige in addition to your powers (hence the name): An elven warrior is OK, but a bladesinger's respected by the People.
- A PrC can be a special path to become exceptionally well in a field of expertise: the Weapon Master focuses on his one chosen weapon, the deepwood sniper is the master of sharp shooting.
- Other PrC's are just unique, like the blighter, which is the ultimate anti-druid, killing nature.
Why take a PrC, or why not?
You might want to take the PrC because of the prestige you gain. ("I'm no normal wizard, I'm a War Wizard of Cormyr!")
You might want to take the PrC because of the Power
("Only +4 Str in Rage? I have Frenzy, and get +10")
You might want to take the PrC because of the unique abilities
("What, you can't do that?")
You might want to take the PrC because it fits your charakter concept
("Why take fighter and wizard levels and just call yourself bladesinger?")
Of course, they aren't always the best choice, and just multiclassing the base classes (or playing one class straight) might be better:
For example, you can style your rogue an assassin, with sneak attack, ranks in hide, move silently, sense motive, spot, listen and so on. While the assassin PrC gives you a death attack and spells, but a rogue will get more skill points and those handy special abilities. You also might want to play an assassin who only slays evil folk, and be a good chap, if an underhanded, backstabbing one, who wouldn't curl the hair of a child. But the PrC requires you to be evil.
While PrC's seldom give some penalties, the have trade-offs, stuff you won't have but had if you stayed with your base class. If you don't need these things, the PrC is OK, but if you want to keep that stuff, keep your fingers from it.
Sometimes, the prerequisites aren't alright for you: like above with the assassin, you may not want to be evil (or not allowed to), or the PrC forces you to multiclass and you don't want (because you don't want that XP penalty for uneven base classes). One of these prerequisites (often hidden) is minimum character level: why take feats you hate to be a verdant lord later? You might not live long enough (because of those feats), or the campaign isn't intendet to get so long.
And then, some PrC's are downright crap.
So, there's no single answer for the question "PrC - yea or nay", for it depends on your charakter, and on your preferences (I like to look for a good PrC which might be funny, and work towards it, but not always.)