Notes on the iPhone doesn't sync with any other application on the Mac or otherwise, which is a limitation of the iPhone. The data are backed up in iTunes along with the rest of the stuff on your phone, but not in a form that would be useful to edit or transfer to another application.
For those reasons, I don't use the Notes application that came with the iPhone. There are several options for backed up notes-type applications in the App Store. The one I use is Notespark, which backs up to the "cloud" and thereby makes its data available from any web browser. It also allows you to share notes with other users. SplashNotes has a desktop application to sync with, if the cloud doesn't suit you. It's actually an outliner, so it is an advanced notes application. And there are other choices too.