Great question! Both quisiera and le gustaría can express "would like," but they have slightly different nuances in Spanish.
Le gustaría is the more common and neutral way to say "would like." It’s often used for wishes, desires, or polite requests, like in the sentence: A tu sobrino le gustaría tener un gato (Your nephew would like to have a cat). This phrasing is more straightforward and typically used when expressing someone's preferences.
Quisiera, which is the imperfect subjunctive of querer, carries a slightly more formal or polite tone and is often used in hypothetical or more polite contexts. It's commonly used for making polite requests or expressing more tentative desires, e.g., Quisiera un poco de agua (I would like some water).
In your example, while Tu sobrino quisiera tener un gato is grammatically correct, it might sound too formal or hypothetical for the given context, which is likely why it wasn't accepted as the correct answer. The sentence is simply stating a straightforward desire, so le gustaría is the preferred choice here.