The book starts, by having a couple use solar sails to just travel through Space for fun. And then, they find a bottle with a message (yes, in Space). And that message is the story, about Ulysse travelling away from Earth, together with a scientist. Ulysse is a reporter.
The start is similar, with them landing on the planet, finding some humans (though they spend a lot more time with them, trying to communicate with them - the first one is Nova, and she is, of course, super-duper attractive), and then being captured by apes.
Then it diverges strongly. I mean, Ulysse and Nova are still captives for some scientists, with Zira taking care of them, and Zaius being her superior (though he is an orang-utan). But instead of whatever the movie was doing, here the apes do experiments. Conditioning, and making them able to do simple tasks. Unlike the movie, Ulysse can't understand the apes, he has to learn the language over some months. Soon, he is able to make it clear to Zira that he is actually smart, and Zira is equally nice like her movie version.
This goes on, until there is a conference, where Zaius (who is barely a character here) wants to show-case how smart Ulysse is, who acts to everyone but Zira, like he is just a pretty smart animal. He uses this to hold a big speech, winning all of society over.
With that, we start what is the basis of the third movie. Ulysse gets clothes, an apartment, and is allowed to move free in ape society. Here, he learns a lot of stuff. He and Cornelius (Ziras husband, also from the movie) are visiting an ancient cave (like the one from the end of the first movie), finding the doll that tells them that there were once humans here.
But aside from that, Nova is pregnant, and the government plans to take the baby away, kill the parents, all that stuff. Generally, people start to get uneasy with Ulysse, and so on.
Ulysse, Nova and the baby (which is shown to grow up as smart as Ulysse) get put on a rocket that is planned for an experiment, but can flee with it. And they travel back to Earth.
They do reach it. But then, we get the ending of the 2001 remake: Apes have developed to be smart, like on the planet they just were on. We cut back to the original ship, with the couple that found the bottle. And learn that they are apes. Despite knowing the movie, this twist actually got me, I thought it was great.
Before fleeing, we learn about the horrible experiments done on humans. One of them seems to recite diary entries, or something, it's really weird and just done, because the author couldn't think of a sensible way. It reads pretty good and disturbing, how a woman describes how her servant gorilla went to meetings at night, and one day demanded that she cleans up, then throwing her out of the bedroom, and then out of her house. Or an animal tamer, who one day got put in the cage by the apes she tamed and from then on had to perform tricks. Or how some scientists tried to inject cancer into an ape, to test a healing method. Except that the apes catch him, and inject him with cancer. Now he is scared, that the healing stuff won't work.
It's a great moment, even though very clumsily done. The humans just get what they did to the apes. It's a pretty harsh critic against the way we treat animals, shown by putting humans in that horrible place.
The book never really explains, why the humans get dumber. One of the diary retellings says that the person starts to feel less intelligent, but that's it. Maybe because they let apes do all the work? It does go into how apes never really got intelligent. They just copied humans. Which is the reason why they are developing so extremely slow, they still only can copy and not think for themselves. And then there are comparisons drawn, to how humans also only copy stuff they don't understand. Like how lawyers just recite texts.
It's a fun idea, but still nonsense, of course. But it's a neat setup for the book.
The book also implies, that Earth and the Ape Planet are different places, and that the development seems to be unavoidable.