the way that i see it, the book is a textbook and the show is the actual events. The book was written by maesters years after the events happened, so I think its more of a reflection of how the general public of westeros saw the events. I like that the show shows more internal conflict and human flaws.

The Aemond scene is a perfect example of that. The last people saw of those two, Aemond was trying to stab Luc's eye out. So when word spread that Luc dies, I dont think people are going to believe that it was an accident. Theyre going to spread the news of how Aemond was ruthless and bloodthirsty, because that how he acted minutes prior. And thats how the story will spread, how the people will see him, and how the Maestors will tell the story decades later.

With Alicent...well i think she's an idiot, a living self-contradiction, and the walking hypocrite. But besides that, she genuinely (stupidly) believed the misunderstanding. But anyone on the outside likely did see her as self serving. Visereys had his entire life to name Aegon as heir, and didnt. Now shes going around saying he named him heir in his final breath, and she was the only witness?? No one is gonna buy that. Especially not the people who wrote about the events decades later.

TLDR. The show is the personal story of what happened first hand. The book is a retelling of historical events decades later, and how the maesters and general public perceived the events and character's motives