Both systems have a restricted set of defined moves that are a large part of the players' mechanical focus. Both also are a major part of leveling up, though Apocalypse World's moves only level you up when highlighted (a player-level choice I never quite grokked) and Chuubo's also gets experience from quests and emotions.
These two ways of setting genre are quite different, though. Since Chuubo's XP actions operate on a narrative level, they can encourage player creativity in terms of narrative structure; for example, in my Halloween one-shot, a player introduced a flashback scene unprompted to use an XP action. Apocalypse World moves, on the other hand, operate on the level of character actions. They can allow for collaborative storytelling when, say, the move asks the player to answer a question or make a choice, but that tends to be more on the detail level than the story structure level. Relatedly, Chuubo's XP actions replace or are orthogonal to other mechanics to accomplish things, and Apocalypse World moves are basically the only mechanic for accomplishing things. I think this makes Apocalypse World moves feel a bit more natural: you make moves based on what the character would do, and if you get XP for it great. In Chuubo's, sometimes it was a bit of a reach to figure out an action that made sense, which can be good for encouraging player steering but can also be less immersive.
I think the vision of genre is quite different between the games, as well. Chuubo's has eight genres in four families and basically four distinct sets of XP actions that define the families, but makes the other moves available according to discretion. At least as our game went, this effectively means you need to be aware of the full list, as one genre's actions doesn't really cover everything you're likely to want to do. Apocalypse World and its various derivatives are completely separate, and it tries to cover everything within its restricted set of moves. It's possible that the Chuubo's in-genre actions would feel more sufficient if I grokked the genre better, but I definitely like Apocalypse World's "do one thing well" focus.
Both have a more specific version that works similarly, as well. Chuubo's has quest XP, which is more specific and only available until the quest is complete; Apocalypse World has playbook-specific moves (the equivalent of class abilities) that drive the feel of a particular playbook and can earn XP in the same way as basic moves.
I guess the main difference here, aside from dicelessness, is the focus on the story level vs the character level, which has a major impact on the feel of the game. I think both approaches are quite interesting, certainly more so than Fate-style lockstep XP or D&D-style combat-centric XP, and I look forward to messing around with them more in the future.