Okay, I assume there are some different reasons for the syncing problem.
I got my notes syncing again by just losing the prefix on my iPhone and renaming the existing notes folder. For those with a similar problem, here's how:
Old situation:
Within OS X Lion I used a Gmail account with notes (IMAP). To avoid a messy layout with the Gmail folders I used [Gmail] as IMAP Path Prefix (found under Accounts, tab Advanced).
Under iOS5, 'Notes' was ticked on under the Gmail account. The result: Apple created a IMAP folder '[Gmail]/Notes' to store all notes.
New situation:
After upgrading to OS X Mountain Lion I ticked 'Notes' under the Gmail account within system prefs, 'Mail, Contacts & Calendars'. The new Notes app remained empty and didn't show my existing notes that were still on my iPhone.
Logging in to www.gmail.com reveiled a new folder created by Mountain Lion: 'Notes'. The existing notes folder was named '[Gmail]/Notes' - because I used the [Gmail] prefix - and contained al the existing notes.
So Mountain Lion syncs with the IMAP folder 'Notes' and the iPhone syncs with the folder '[Gmail]/Notes'
In Lion the Mail settings were used for both Mail and Notes. So the IMAP prefix applied to both.
In Mountain Lion, the IMAP prefix only applies to Mail, not to Notes.
New situation, fixed:
As Mountain Lion can only use a folder called 'Notes' for syncing, I had to change the folder my iPhone was syncing with: '[Gmail]/Notes'. I removed the newly created folder 'Notes' via www.gmail.com under the label settings and renamed label '[Gmail]/Notes' to 'Notes'. After that I removed the IMAP prefix '[Gmail]' from my iPhone Gmail (advanced) account settings, making the iPhone sync notes with the folder 'Notes' instead of '[Gmail]/Notes'. That's it.
Btw, I maintained the use of the prefix within Apple Mail under Mountain Lion as that is totally separated from the Notes section.
Hope this helps.