What changes were made in Princess Kaguya from the original Tale of the Bamboo Cutter story?

There is no definitive list of changes from an official source, however here are some related interview questions posed to director Isao Takahata:

Q: Did adapting a folk tale like the Bamboo Cutter mean that you could apply your own personal themes? It seems to me that Princess Kaguya is about the cyclical nature of life and the brevity of happiness - things that, if I'm not mistaken, weren't in the original folk story.

A: I agree with your point. The songs sung in the film were meant to give praise to the Earth, this star where we live, for its marvellous cycle of life. And my effort was to show how The Princess Kaguya fully enjoyed the richness of life on Earth that is filled with colour and all sorts of creatures as well as the joy of living and of forming emotional bonds.

--- from a 2015 Interview with denofgeek.com

and

Q: Did your original idea include this unique visual style or was it more of an idea of how to interpret the story for the big screen?

A: It certainly was the story. In the original story, it's very hard to understand the motivations of the main female character [Princess Kaguya]. So that makes the story mysterious and interesting. But I thought if I could inject whywhy she had to return to the moon, then it would be able to be told as a [cinematic] story. But, of course that was not the style, because 50 years ago I wouldn't have thought of using this kind of drawing style to show the film.

Q: Since this fairy tale story well-known in Japan [as ???? Taketori Monogatari], did you feel more of a pressure to create something that was unique, and something that was different than what everyone was already familiar with?

A: So my aspiration, I didn't feel pressure, but I felt that what I wanted was for people who watch the film to say, "Oh, was this what was really behind the story of Princess Kaguya?"

Q: In Japanese culture, has this story been more of an influence on society or more in reflection of the way society has evolved, especially with where it's at today?

A: I haven't really regarded the story in either of those ways. But, for example, in the story, the five suitors, in the original story they are told by the Princess Kaguya to bring something back, each of them, something that only someone who had had education in Chinese Classics would know. So she was supposed to have known all this, right?

But in the film, what I did was have her say, "Well, you've praised me with that kind of metaphor or simile. So then bring back what you think my value is." So it's not that she had to have a Chinese scholarship or anything. So the meaning has changed then. It wasn't just to make it really troublesome to those suitors to cause a lot of trouble. It has more of a modern appeal to the modern audience, I think. "If you think of me in that way, then bring me those real things that you've compared me to." So she is rebelling against being so objectified by these suitors. And that's a very modern sensibility.

--- from a 2014 interview with firstshowing.net

To me it sounds like the largest changes made to the original story were made to better illustrate the motivations of the protagonist.