Continuity mistake: When Harry is viewing Snape's memories, he witnesses a scene on Platform 9 3/4 between his mother and aunt Pentunia, in which Petunia tugs her hand out of Lily's grasp. Her hand is then back in Lily's and she tugs it out again.
Other mistake: Ron gives Harry a book for his birthday, which he says Fred and George gave him last year on his birthday. He says he wished he had it when he was with Lavender. However, Ron was with her on his birthday last year.
Factual error: Harry et al. are captured by Death Eaters in March (it's mentioned part way through chapter 22 that Ron was unable to find Potterwatch on the radio until one night in March; they are captured shortly after the program ends). Greyback and the other Snatchers take Harry and his friends to Malfoy Manor, and Narcissa states that, if this is indeed Harry Potter, her son, Draco, will recognize him. She says he is home on Easter break. However, this is 1998 (Harry turned 17 near the beginning of the book, which would've made it 1997, and this was the following March), and Easter was April 12th that year. Draco would still be at school.
Continuity mistake: When Harry talks to Griphook about breaking into Gringotts, Griphook has the Sword of Gryffindor in his hands. When Harry stands up to leave a few moment later, the sword is now beside Griphook's bed.
Answer: Voldemort more than likely saw Lily as a valuable bargaining chip. JK Rowling revealed in an interview that Voldemort always planned on killing James (which is why his death did not count as a sacrificial love). The reason Voldemort agreed to Snape's pleas to offer Lily a chance to live, is because he rewards his followers (always at a cost, however). Voldemort likely agreed only because he probably thought he could use Lily as leverage to ensure Snape was at his mercy.