Game of Thrones made several changes in adapting Daenerys Targaryen from George R.R. Martin's A Song of Ice and Fire books, but one part cut from her origins and backstory was particularly key to understanding her journey. The HBO adaptation kept the broad strokes of Daenerys' story in tact, charting her rise to power in Essos and filling in some gaps of her past, such as the fall of her father, the Mad King Aerys II Targaryen, and her birth on Dragonstone that saw her labelled Daenerys Stormborn. Somewhat inevitably, though, changes were made along the way as well.
Even Daenerys' appearance in Game of Thrones is different to the books, with the character most noticably lacking the purple eyes her on-page counterpart has (due to the uncomfortable contacts that were required). So, too, will more changes likely become apparent when Martin finishes the series, as Dany's story in The Winds of Winter and A Dream of Spring should take things in some different directions, even if some of the controversial aspects of her "Mad Queen" arc presumably will happen in some form. Daenerys' future is very much informed by her past, and one of the important aspects of that in the books didn't make it into the show.
In the A Song of Ice and Fire novels, Daenerys sometimes thinks back to a house with a red door, believed to be in Braavos, which had a lemon tree growing outside of it. This is where she and her brother, Viserys, lived for a time with Ser Willem Darry, a knight loyal to House Targaryen, who cared for the children during their early years in exile. While they eventually had to leave after Darry's death, it was the closest thing Daenerys had ever known to a true home, and that's what makes it so important. Game of Thrones focused on Daenerys' desire to take the Iron Throne as her birthright, but the house with the red door shows the other side of that. It isn't just about power or even about what's rightfully hers, but it highlights that part of Daenerys is still the scared little girl who just wants to find her way back to her home and family, adding more layers of depth and vulnerability to her story, while giving her quest for a throne other motivations. That it was cut speaks to the show's different format, but also a different presentation of Daenerys too.
One of the biggest challenges with adapting the A Song of Ice and Fire books is the point-of-view format they are written in, and many of Daenerys' best moments in particular come from her internalized thoughts and feelings. Her dreams of and longing for the house with the red door form part of that, so don't make for an easy translation to the screen. That's especially true when so much has to be cut in order to fit into 10 episodes a season, and since the red door doesn't change Daenerys' narrative trajectory (or at least, hasn't so far) then again it's easy to see why it was cut for the sake of expediency with regards to her backstory, even if it would have made things much stronger with it included.
Similarly, Game of Thrones often chose to focus on Daenerys as the strong would-be Queen who would take what is hers with fire and blood. In effect, the show's character was often more khaleesi than it was Daenerys; the powerful ruler, not the more vulnerable, lonely girl who longs for home as much as she does the Iron Throne. Becoming Queen does hold the allure of power and taking what Dany perceives as being hers by right, but it's also what she think is best her chance at going "home" and re-connecting to her family, since the red door, Willem Darry, and even Viserys are all lost. Game of Thrones wanted to underline the importance of the Iron Throne as Dany's goal, which makes sense given how much it factors into her entire arc and ultimate endgame, but it misses some of the nuance of that. For instance, Game of Thrones season 8, episode 2 has Daenerys say: "All my life I've known one goal. The Iron Throne. Taking it back from the people who destroyed my family." And yet, in A Game of Thrones, Martin writes: "All Daenerys wanted back was the big house with the red door, the lemon tree outside her window, the childhood she had never known."
Those two quotes perfectly highlight the difference in approach to the character, who goes on the same path plot-wise, but with contrasting drives and complexities. That isn't to say Daenerys' arc in Game of Thrones was bad - for the most part, it was well done, and Emilia Clarke did a stunning job in the final season in particular. But given how much of the controversy around her descent into "madness" stemmed from not showing enough nuance or development of her feelings, then it's easy to imagine how much stronger it would've been - and will be in the books - if her desire for the throne had been mixed more with her sadness and dreams of being home and safe once again.