Hi Iris,
Both phrasal verbs *can* be used to talk about the personal movement of the body. "She pulled away from his embrace and looked into his eyes," and "The children nervously moved away from the barking dog."
But "pull away" has the figurative meaning of "create emotional distance between two people who were/are/could be in a relationship", and "move away" doesn't really have this meaning. So, "Claire sensed that Jimmy was pulling away from her when he started waking up before her to leave for work."
But "move away" is the phrasal verb for leaving people/places/ideas behind when you move from A to B.