I don’t think the same person wrote the note & letter; probably just the same PA wrote both props. Personally, I think Neal sent the note based on how he and AWB used to communicate. For reasons unknown, he’s not able to get Emma himself but he either knows what is happening and is unable to go himself OR he knew what was about to happen and sent the note to Hook via a dove before it was too late. Neal knows that Hook can travel from SB to NY via the Jolly Roger so he sent the note to Hook to get Emma & save everyone in SB.