I think FitD is more about bouncing off a world that already exists outside your characters, and PbtA is more about building out a world around the characters as move results tell you things about it.  There are other differences that mostly amount to sliders of "actor vs director" and other such meta-game concepts, but the nature of the setting and the players'/characters' place in it is a big one.

Hey World Champ!

Let's talk Official Difference vs System Difference.

To understand PbtA and FitD you can look at the legal aspects. Vincent and Meguey (who own PbtA) have said before that what PbtA officially (aka legally) means is that you feel your game was inspired enough by Apocalypse World that you feel like putting the label on. There are no other requirements. If you actually copy any Vincent/Meg words you have to get their permission to re-use them, but that's normal for any direct copy of text. Here's the official stance: http://apocalypse-world.com/pbta/policy

Now the community (system difference) may mean different things. I have no authority, and I don't want you to read the following text as if I'm saying this from a place of official canon, but it's just an observation on my part. Usually when someone says they have a PbtA game you can expect any of the following:

• Moves. You'll have a conversation which when a certain fictional event happens resorts to mechanics to resolve it.
• Partial Success. Many PbtA games roll 2d6 and add a stat (usually -1 to +3) to resolve moves, with some moves being full successes (10+) where the character performing the move gets what they want with no drawbacks, 7-9 where they succeed at a cost, or 6- where it's a miss, and the GM will introduce a twist or complication, though not always as a detriment or direct consequence of the action the player undertook.
• GM Rules. Apocalypse world successors often have Principles and Agendas which tell the GM (often labeled the MC aka Master of Ceremonies) what to keep in mind, and what to narrate outcomes in service of while running the game. They also have a list of actions to take on misses to help shape the fiction of their outcomes.
• Playbooks. Many PbtA games are GM facing. The players are handed character sheets with all the information needed to make characters and play them self contained on them.

These is not an exhaustive list. And there are PbtA games that break these rules (a cool example is Undying - in which you play vampires. It's diceless, and resolves many moves with systems of bidding blood). Here's a big list of PbtA games you can look at: http://apocalypse-world.com/pbta/

Let's talk about Blades in the Dark! Forged in the Dark (FitD) uses the Blades engine. Legally it's CC-A 3.0 Unported (https://bladesinthedark.com/licensing). Harper went one step further than the bakers. He put up an SRD (System Reference Document) which you can re-use and copy directly, as long as you keep the CC-A note and the FitD logo on the game. Consequently many FitD hacks have more standardised text that replicates directly.

FitD is itself a PbtA game! (shock! Awe!) There's some deep cut discussion about how it's core system is actually moves, but rather than getting all deep and game designery let me do a high level overview of things you might expect.

• Fiction Matters. Outcomes and consequences change based on the scenario. Whereas in something like DnD a weapon does 1d10 damage no matter what, here how much harm you take depends on the circumstances. Are you in a controlled position (in cover, covered in armor)? Then you probably won't die from a musket shot. Are you making a mad dash under fire? Consequences might be dire. 
• Player Authorship. Resistance rolls reduce consequences, and can change fiction AFTER the roll. Players have a limited pool of stress to do so, but GMs describe the worst that could happen, and the players say no, and can change outcomes to some degree.
• Phases. Play often takes place in phases. Often there's a zoomed in mission phase where folks take actions moment-to-moment, and then a zoomed out downtime phase that's different and more structured where you take recuperation actions, and do some free-play.
• Flashbacks. Blades prizes speed, and again that player authorship. It encourages you to dive in head-first trusting that your competent characters can use mechanics to retcon the story and produce preparation without having to meticulously play through it.

Again, some of this changes in hacks (Band of Blades doesn't keep fidelity to a single character, but a pool of characters that change who plays them mission to mission for example). You'll also see PbtA roots in the system (the core moves are just that, moves. Character sheets are similar to playbooks, though Character Creation is often separate. There are GM Principles and Agendas. There are partial successes, though blades uses a die pool).

Hope that helps friend!

To tack on to what @strasa said, you can view Blades in the Dark's core resolution mechanic as a on the fly custom move generator with the back and forth about position and effect.

And I also would not discount how heavy of an influence I feel Burning Wheel has been on Blades and thus subsequent FitD games. Its got the core of reflecting on "beliefs" as a major drive, two of the three xp triggers for characters are about that tension.
It is much less obvious than the more appearance-leaning bits like playbooks, which I feel are very much smoke and mirrors for accessibility's sake (like, nothing actually breaks if you mix and match different xp triggers, special abilities, items, contacts, gather info questions... it would just be a lot to take in if it came presented in that modular way).

Then there is also the thing where most PbtA games tend to revolve around 8-12 session campaigns and FitD easily goes for 20+. One is not better than the other but its a major difference in my mind.

For me, the real crucial different between PbtA and FitD is that FitD tracks some mechanics for the entire group. In Blades, you have your terf and your heat, in Scum and Villainy, you have your ship. Meanwhile, PbtA only tracks mechanics personally. Sure, the town you're in may have some scarcities, but that really only matters for the people in charge and not for the wandering gunlugger. This means that FitD games tend to focus on intra-group dynamics to a degree that PbtA does not have to.