Lilly is in fact Roy's mother. Its briefly addressed in the book that Lilly was married off at thirteen and gave birth to Roy just after she turned fourteen. And then her husband died and it was just her and Roy after that.
In the book when Lilly is trying to coerce Roy into giving her the money, she tells him that while they may be criminals they've always kept their private lives fairly straight and that there have been certain things they just wouldnt do. Roy says that it would have complicated things. After Lilly begins to flirt with him at this point and she sees that he is responding, she blames herself for this because of her age and how she had raised him. So, in the book, there is this sexual tension between them but they havent ever done anyting about it.
Also, the book states that Roy left home because he couldnt afford to go to college after high school and he wanted to grift himself. He wanted to prove himself to Lilly as an independent man who could do without her help instead of the weakling he always seemed to be as a child. Lilly just wanted him to get an education and even offered to pay for it. They were just working toward different goals and that's what I think initially caused a rift between them.