What is the conflict between the mother and narrator in "Two Kinds" by Amy Tan?

The conflict between Jing Mei and her mother stems from the latter's need to turn her daughter into a prodigy of sorts.
As the text tells us, Jing Mei's mother is focused on her mission to make Jing Mei a piano prodigy. This becomes a point of conflict between the two. While Jing Mei's mother thinks making her daughter into a star (on par with Shirley Temple) is the right thing to do, Jing Mei isn't so sure. She feels pressured by her mother and resents the need to put on an obedient front before relatives and the larger Chinese-American community.
Jing Mei just wants to be herself and be allowed to choose her own path in life. Thus, her mother's simultaneous need to keep up the family's reputation and to compel unquestioning obedience is seen as oppressing to her. In the end, Jing Mei rebels by playing horribly in a piano recital; in the aftermath of the disastrous performance, Jing Mei engages in an emotionally charged argument with her mother. She accuses her mother of not accepting her and expecting her to be something she's not.
Jing Mei's final words during the argument decide the result of the conflict, but it proves to be an unsatisfying victory. By cruelly referencing her mother's dead babies from her first marriage, Jing Mei manages to hurt her mother as well as to dispirit her. She reports the piano lessons stopped soon after, and she stopped playing the piano entirely. It is years before Jing Mei realizes the faith her mother had in her in that conflict-ridden period of her youth.