Did you realize that every time you create a new note, it seems to always access the internet even though you do not have any apps running in the background?
Note the network activity loading indicator on the statusbar.
Apparently, if you set up your email client (assume the mail server support IMAP), in my case gmail, it will automatically turn on syncing for notes.
So each time you modify your notes (or create/delete notes) in the default Notes app, it will save this changes to your mail server.
For example, if you are using gmail, the Notes app will create a Notes label and all the notes will be saved as entries having the Notes label. However, the process is one-directional, so you can’t modify the notes in your Gmail.
Of course, if you prefer not accessing the net and not syncing the notes, you can turn off this features in the Settings app under every email acount.