First of all, bare in mind, the following is my personal opinion. What's also to point out is that the book is written as a screenplay, meaning the West End production of "Harry Potter and the Cursed Child"? That's it! Left to your imagination. Some minor spoilers can occur, so you are warned!
"The Eighth Story. Ninenteen Years Later." (Bookcover)
That's what it says on the back of the book, which generally sums up the story. In a proper Potter manner, it is his son, not Harry himself, who discovers a mystery this time, together with his intelligent outsider friend, sounding very much like Hermione there. Both boys are carrying an inherited package and try to solve this riddle they are stumbling across together. Parallely, Harry and his friends are finding out about the story themselves but it appears that they are always a bit too late for the two boys.. Rings a bell? Hello there Mr. Dumbledore, sir. Many characters of the former books are brought back and there are only few additions made, mainly on the part of the current students of Hogwarts (for obvious reasons).
One thing I've always loved about J.K.Rowling was that every detail in her stories is not just made up - there are actual stories behind all names and places that personify the person/place they stand for and pretty much sum up their character without revealing too much about it to people that are not familiar especially with Greek mythology as well as the signs of the zodiac.
She kept true in this part, as to asigning a lot of traits already by naming people. I'm very happy about that because I truly believe J.K. Rowling to be brilliant for this reason - everything has a deeper meaning! EVERYTHING!
To be honest, the nicest way to put it, is that it builds up. I never thought I'd have to say that, but most of the first Act is just plain boring. It's not surprising that a book needs introduction, but under the given facts - a screenplay, only 330 pages (remember, that does not last as long as a novel of 300 pages would), I do not like the fact that in the first third of the book nothing significant or catching happens.
If all, I was bewildered by the part with the trolley witch, which, for me, did not sound very authentic, but rather as to put some action into this part of the story. It appears this bit is completely disconnected from the rest of the book and has no further meaning, very un-Rowling-ish.
Futhermore, unluckily, I did not bond with any of the characters in the beginning. Everyone had their own, unlikeable traits that made it hard for me to like any of them. This improved over time, but compared to the other Harry Potter books, it was truly disappointing and I was not sure if I would really read the whole book, in the beginning.
The rest of the story is a lot of What-ifs and I really like the way she goes through different scenarios if only one tiny part of the plot had changed.
Especially in the end, I was waiting for that one more, dramatic, unforeseen, Rowling-esk death of someone, but (luckily) no one (new) died. Instead, the end of the story is very impressive and very much like the other Harry Potter books, very moving, painful but beautiful and everything is more or less falling into place.
Unfortunately, my impression was that you could foresee a lot of the plot. In the end, it was not revolutionary or a significant twist that gave a whole new meaning to it all (remember Snape's story - a masterstroke of her's btw). It was, at times, almost a simple retelling of the original parts maybe from a different perspective. Saying it was rehashed would probably go to far, but I was missing some originality and innovation.
Screenplay vs. Novel
Comparing the former novels she has written with the screenplays she wrote recently (the other one was for the Fantastic Beasts film), I would say I prefer the novels over the screenplay.
Firstly, they have a lot more detail, which, for obvious reasons, cannot be transferred to the screenplays. Nevertheless, I believe that Rowling's uniqueness is mainly coming from the way she connects everything, but also the beautiful detailed descriptions she gives, which are lost as soon as the novel turns into a screenplay.
I do understand that the screenplays were her reaction to people complaining about the films being so different and much plainer than the books. But I doubt fans hoped for an assimilation in this direction. Much rather, I believe, they hoped for longer films but with a greater amount of detail, not books that lack depth, to put it like that.
Secondly, I feel like fans are being cheated of another excitement. Usually you would first read the book and then, sometime later, go to the cinema and watch a film made on the basis of this book. You can look forward to two things and you do not know how things will turn out. When you have a screenplay on the other hand, you very much know what is going to happen, as there is not such a great difference between the book and the film/play.
A nice try and new Harry adventure fans definitely looked forward to. Unfortunately, it lacks depth and uniqueness, making it appear a lot less appealing than the other books she has written. The story takes some time but evolves to be quite interesting, though struggling to be fully catchy as well as emotional in a way that people were moved by it.
Welcome to the club. Sit back, drink a cup of tea and enjoy what comes out of my crazy little head :D