This is a review that compares the actual play onstage in London with the script in book form (with thoughts on the general story as well). Spoilers are hidden. 
 I watched the play in London on 4 August (Part One) and 5 August (Part Two) and finished the book a few days later - so I had everything from the play fresh in mind. 
 Firstly I want to say that the play was beautifully done. All the illusions and effects used to achieve magic was ingenious, to say the least. The audience was wow'ed over and over again and we all gasped and 'oh!'ed and clapped in unison when something particularly brilliant was pulled off. This is obviously something the script lacks; as it is not a novel, there are no in-depth descriptions of how a spell works or how something magical happens. This means that your imagination is working over-time, relying on everything it has learned earlier from the books and films to fill out the gaps. My guess is that this is easier for some people than others. 
 Something else which I've seen in several reviews already; the fanfiction vibe this script has. It is true, it does. But in the theatre, this is not obvious at all. This script was made for the stage, and theatre is a very different form of entertainment from screen and books - it relies heavily on its audience and the ability to constantly keep up the pace and the wow-factor. Thus an overload of jokes and over-the-top plot points have been incorporated, making it seem a little far-fetched, at least on paper. I can tell those of you who haven't seen the play that this all works rather well onstage; it keeps the audience going and we all laughed along with the jokes (especially Scorpius') and sat on the edge of our seats when something exciting was happening. 
 In book form all this falls a little flat - again we miss out on the effects onstage that supports many of the plot points that seem far-fetched on paper. The jokes seem one too many. The characters seem like ghosts of themselves, because we lack descriptions of their tone, their facial expressions, their gestures. All that we get onstage, and though some characters didn't seem like the perfect casting, most of them did a brilliant job - especially Harry was absolutely perfect and sounded and acted exactly how I would imagine 'our' Harry as an adult would. 
 Plot discussion time: again addressing the fanfiction vibe. 
 I have one part of the plot that I cannot cope with and I will not accept it as being true: (view spoiler)[Delphi being Voldemort's daugther by Bellatrix. With a character like Voldemort who is so possessed by power and who cannot love, and who is indifferent to others' desires and devotion, it just doesn't add up. I could have accepted if they made her a child of Bellatrix' who had inherited her mother's obsession with the Dark Lord. My boyfriend argued that Voldemort might want an heir, but I don't believe that either, because Voldemort wants to live forever and thus there is no reason for an heir (hide spoiler)]. I stand by this and my claim that this would never have happened in a novel by JK. Therefore I have decided to pretend it didn't happen. Sorry. 
 The (view spoiler)[Time-Turner (hide spoiler)] plot has also received some hits from fans. Claims of inconsistencies and the likes. I agree that this is an example of plot points that seem far-fetched and over-the-top, but I also believe that JK could have worked wonders with this in an actual novel and I am still and of course bummed that this will not happen. However, theatre doesn't have much time and space to work with and this is already a 5 hour play in two parts! 
 Albus and Scorpius are the stars of this script/play and are both beautifully written and acted. The play itself was brilliant and entertaining and quite well done. The script falls a little flat without the actors, sets and effects to support it. Personally I don't think it should exist in book form, mainly because it will ruin this new story for many fans. But I also realise that those who cannot see the play would want to experience it in another form, and I would probably feel the same way if I hadn't been able to see it either. The play would get 5 stars from me, and the book gets 3. 
 All in all I have difficulties viewing it as a continuation of the books. To me it's more of an addition to the universe, much like Fantastic Beasts is going to be. 
 If you read all this, thank you! I realise it is quite long ;)