One other curious thing, for me, about PbtA/FitD games, is that I'll happily grab a playbook and make up a character, whereas D&D classes have always irked me. Maybe it's that playbooks—which many think of as classes—are pretty minimal compared to D&D classes, which fill in a lot of stuff I would rather do, meaning I'm either shoehorning my concept or doing extra work to reflavor (which isn't even always possible).
What I'm about to post is conjecture. Not utterly wild, random conjecture; but not grounded in systematic study of the full range of cases.
I think there is a tenable argument that the D&D class system peaked somewhere in the mid- to late 70s. At that point, and as Gygax said in his PHB, choice of class reflected a players' choice as to
how they think they can best meet the challenges posed by the game. Classes were suites of abilities coupled with limitations, perhaps severe ones (if playing a cleric or MU or similar class) or else were an (unfortunately weak) utility kit for dungeon exploration (if a thief) or a degree of survivability that then made play-via-fictional-positioning possible (if a fighter).
But why can't my class be a crafter? Because making swords and/or pottery isn't a way of meeting the challenges posed by the game!
The orientation of a lot of D&D play has changed pretty dramatically since then - "story" or characterisation seem to be widely valued - but the legacy of those classes lives on. In some cases it even rebounds back to infect or shape the underlying fiction, which is just the pits!
(I think 4e is something of a successful reversion back to Gygax but without dumping the theme stuff. But that's another story.)
In the case of a PbtA playbook, on the other hand, unless it's at the weaker end of the spectrum
@chaochou has described, it should plug you right into the core play of the game. If you build your PC using the playbook but still feel stranded or ill-shaped vis-a-vis the setting or starting situation, then you know you picked up one of the duds!