1.
Ron is a goofball. He's pure comic relief and this is achieved by making him an emasculated idiot. I say emasculated because he is so whipped. Hermione wears the pants in the relationship. Fine. But Ron was never a dog following her around trying to make her happy with cheesy jokes and generally just being there in the background. He was assertive. He was proactive. She admired that about him. The Ron of
The Cursed Child is not the Ron of the novels and it is almost painful to see him relegated to such a minor role. As for the idiot part, there's a scene in which he gets his wand out and holds it the wrong way. The only time he ever had trouble with his wand in the novels was when the Whomping Willow broke it.
2.
Voldemort has a daughter. If there's anything that makes
The Cursed Child come across as glorified fanfiction this is it. This contradicts established canon and Voldemort's character. First, Voldemort's weakness is his total inability to love. He never has friends, he never trusts anybody. Voldemort believes that other people are completely other to him, that everyone is expendable. The idea that this guy would sleep with Bellatrix just doesn't fit. If we assume he slept with her because he cared about her then we hit a wall because his entire character is defined by his inability to feel anything other than hatred for others. He would never have slept with her because he is not capable of feeling anything like love, lust, or even empathy towards another person. If we assume he slept with her just to create an heir, that there was no emotion in the act whatsoever, we also hit a wall because Voldemort has no reason to believe he could be defeated until he hears about the horcrux being stolen from Gringotts. That doesn't give him enough time to get Bellatrix pregnant. The idea of not conquering the world himself also wouldn't occur to him as eternal life is his ultimate goal. Leaving a legacy for his child to secure wouldn't enter his mind.
Second, if Delphini was born just before the Battle of Hogwarts then Bellatrix Lestrange would have been pregnant throughout
Deathly Hallows. This simply isn't the case. I find it hard to believe no one would talk about such a thing at any point. Bellatrix pregnant with Voldemort's baby would have indulged in such an
honour for everything she was worth. Rubbing her belly, waxing lyrical about how the Dark Lord had chosen
her, proving how pure her blood was. Nothing like that is going on because Bellatrix wasn't pregnant. Her giving birth simply doesn't fit the established canon. Of course, retcons are fairly common in fiction, but the reason why something like this is an issue is because if you go back to read
Deathly Hallows you will be aware of how it doesn't fit, how it is now
wrong. And it will make you enjoy the story less.
3.
The Fidelius Charm. In the last Act of the play, Albus and Scorpius go back to 1981 and see Lily and James with an infant Harry. This goes against established canon because they were protected by the Fidelius Charm, a piece of magic that makes something completely invisible and undetectable to everybody that has not been told the secret. This would make it impossible for Albus, Scorpius, and Delphini to find the Potters. And yet. It's possible that they could see through the Charm because they all knew that Harry and his parents lived in Godric's Hollow, but it's unlikely that they knew the exact address (which was required for Harry to see Grimmauld Place in
Order of the Phoenix) and the Charm had been inactive for about 30 years when they were told, making it highly unlikely, if not outright impossible, for them to see through it.
4.
Snape is alive in one of the alternate futures. Albus and Scorpius change the past and when they return to the future Scorpius finds himself in a world where Voldemort killed Harry and took over the world. Snape is still alive in this future. The problem is that Snape would have died no matter what because he was killed so Voldemort could become master of the Elder Wand. Harry dying in his duel with Voldemort happened after Snape was killed, meaning that Snape would still definitely be dead regardless of Voldemort killing Harry. And yet.
5.
Loads of little things. Nit-picky stuff like Harry still having the Marauder's Map despite
JK saying that his eldest son stole it from him. Professor McGonagall being the headmistress despite
JK telling us she wasn't. Harry, Ron, and Hermione not realising someone was making Polyjuice Potion when told of what ingredients had been stolen from the Hogwarts storeroom. The gang also talking about using Polyjuice Potion on a whim as though they forgot it takes a month to brew.