I believe Beyond corrects many of the major mistakes made in Indigo Prophecy and Heavy Rain, and it does this largely by substantially improving the quality of the writing, scenarios, and characterization.
The most major evidence of the quality of the writing being improved is the fact that it is written in English. Indigo Prophecy and Heavy Rain are filled with ESLisms or awkward translations. Y2Kev has mentioned it before, but one of the most baffling in Heavy Rain is when a garbage dump is called a "wasteland". It comes through in pretty much every sentence. It's especially bad in Heavy Rain because the actors chosen are clearly not native English speakers and virtually every performance is poorly directed and contains heavy accenting or awkward pronunciations. This is not the case in Beyond. There's definitely an argument to be made that some of the subject matter was trite, but the dialogue at least has a relatively natural quality to it and the performances, even the ones by supporting characters, appear to be English-speaking actors with an English voice director being given appropriate feedback.
The second biggest fix is, compared to Heavy Rain, Beyond appears to have not had as much cut content and as a result the story flows from beginning to end much better than Heavy Rain. Very major things were removed from Heavy Rain. Major components of Madison's and Ethan's plotlines, actually. So there are things in the game that don't really make sense and are evidence of those things left over. As well, the central reveal associated with Shelby is generally considered to have been handled poorly. Nothing like this happens in Beyond. Probably the weakest overarching element of Beyond's plotting is the connection between Jodie and Ryan, which seems a little forced in the finished product. But this is definitely a minor issue as compared to Heavy Rain. I'd also add that several endings emphasize the importance of Zoey in a way that the game probably doesn't support--whether this is intended to be a tease or is a result of cut content is unclear. But this isn't leading to people saying "The game's story doesn't make sense".
As a result, I feel Beyond comes closer than ever to achieving the goal that David Cage set out for it. I think that's what makes the game a pretty unqualified success on its stated goals. As a matter of personal appeal, I think I liked Beyond's story more than Heavy Rain's. I do enjoy crime stories, and I thought Heavy Rain's musings on family were occasionally interesting, but I think Beyond was a more personal, contemplative story. I am surprised that the sort of audience that likes art-games, personal games, emancipatory games that try to tell personal or human stories did not like Beyond more. I thought it did a fairly good job of establishing the trajectory of a life, I thought characterization was fairly good, I thought it was able to achieve genuine emotion in parts without being too maudlin. Now, some people don't like experience games, because they're looking for more skill-driven, score-attack type gameplay, and I think that's a valid approach to games and if you're someone who likes shmups or fighters it's pretty obvious that David Cage is not really making games tailored to you. But within the confines of the kind of games it's competing against, the sorts of similar experiences, I think it's a success. A strong success.
From the non-story, gameplay side of things, I think Heavy Rain substantially corrected the major issues with Indigo Prophecy. Indigo Prophecy had very lengthy scenes of mashing Simon Says QTEs. Heavy Rain corrects this by shortening the length of action scenes and by remapping controls to make more sense contextually, so that the action on the screen reflects the buttons you're being asked to press. This makes the scenes far more enjoyable in a big way, especially scenes that have you almost playing twister with your hands. Indigo Prophecy had frequent permanent fail states attached to QTEs, where if you pressed the wrong button, you game-overed. Combined with the length of these scene, this was very frustrating. Heavy Rain fixes it by threading in failure into the narrative and making it add some elements of diversity to the story in terms of how it plays out. This makes failure feel more deserved and reasonable, giving the game a level of challenge without the frustration of having to repeat a rote and boring section.
I can't say that Beyond builds on these improvements, and ultimately I actually think the stick-dodging is weaker for action scenes than Heavy Rain's button prompts. I think it's weaker because it's difficult to ascertain the direction of movement sometimes (IE Jodie is dodging left but moving her arms right, or she's sort of dodging in two directions). Also there are fewer story junctions where failure has long-term ramifications or causes the player to miss scenes, so it's a little less rewarding.
In summary CAEG IS BAD RITER WER DA GAMEPLAY I HOPE QD GO BUNK RAPT NON GAME LOL GO BACK TO CEL PHON