I spell it Lily like the flower. I'm used to seeing Lily rather than Lilly. I don't mind Lilly as a nickname for Lillian, but I don't care for it on its own. Lilly seems too much like a nickname to me. Lily, on the other hand, seems more substantial because of the word.
I never knew that Lily or Lilly could ever be on its own. I thought that they both were nicknames for Lillian. I've only met Lillians, never straight Lilys.
That's what I mean. All of the Lilly's you know being nick names for Lillian seems odd. I have seen more plain Lilly and Lily than I have seen Lillian.