Hi junxiang,
It appears you are using two different "versions" of the notes "process". Prior to the introduction of Mountain Lion, (and I'm not an expert on this so forgive me if I misspeak), apparently the Notes App that was on IOS devices was synced to a Mac via Mail.
Notes from an IOS device are now synced with a Mac using iCloud, not mail. You can email a note to your Mail account from your iPhone 5 by clicking on the little "share" button at the bottom of the note, but it won't automatically go to your mail.
The "Sync Notes" option in iTunes is for syncing notes between devices that share that iTunes account. However, if you do go with the iCloud option for syncing notes between your devices, then you should turn off the iTunes notes sync or you will get duplicates.
Bottom line, unless you manually "share" your notes with your email account, I do not believe they will sync through mail anymore.
Also, you should think about freeing up some memory on your iPhone, or you undoubtedly will run into issues eventually. It looks pretty maxed out! Maybe sync your photos to iPhoto on your Mac and delete some of them? (just a thought).
One other thing. If you upgrade to Mountain Lion, then you will have the same Notes app on your Mac that you do on your iPhone. Then with either iCloud or via iTunes, you will be able to sync your notes between the Mac and the iPhone.
Here is a link with the current Notes sync info from Apple:
http://support.apple.com/kb/HT4191
Hope this helps!
Cheers,
GB
