I think Lily's face says it all. She is kinda scared of cats. Something about Miss Socks Lucyfur (about to be Sweet 16) bopping Lily when Lily was a baby and she hasn't been very happy with cats since. However, Lily's toleration level of Luna is high compared to her other two reactions: run after kitty (all but Socks) and cower (Socks).
Lily Jayne & Luna Lovegood