A note that is added to an order by the buyer is something that you see, as the seller, and are supposed to take into consideration when packing and making the product, such as colour, specific packaging request and so on.
If you are the seller, however, and you are looking at an invoice, "add a private note" just means that ONLY YOU can see it, not the buyer. I use this space to update order statuses to keep Etsy happy in case something goes wrong with a complicated order, such as if I have to contact the buyer to ask a question, and I need to wait for their response, and I make sure to date each note.