My introduction to role-playing games came from Dungeons and Dragons, the red box. Of course I was captivated. You could be a fighter, a magic-user, even an elf. What an awesome concept!
A few years later, I played another pen-and-paper game called Champions. This was a superhero game, and the amazing thing about it was that you could be any kind of hero you wanted. There were no rigid character classes; instead, you bought your hero's abilities with character points. There seemed to be no limit to what type of character you could make.
These days, most people experience role-playing games on a computer. But the old "fixed classes vs point-buy" debate still goes on. Some games make you pick from a menu of predefined classes, while others let you heavily customize your class through skill trees or even build your own class from scratch.
Although the number of choices in a point buy system seems limitless, in reality, dedicated players quickly figure out which builds are the best ones. As these builds get published on wikis and message boards, the community converges on a very small set of viable character types. Meanwhile, newbies faced with a near-infinite set of choices shrug and just pick something, and that something tends to have sub-par effectiveness.
With fixed classes, the game designers carefully curate the types of experiences that the players will have. Designers make each class unique, and they carefully balance them to achieve the target gameplay. By contrast, the designers of a point-buy system essentially leave the set of classes up to the vagaries of emergence. And so I think predefined classes are better.