I started off with the Evernote option, but Evernote sync in the Note 2's S Notes failed. Has worked numerous times before. Could be a temporary issue. Since I didn't have Evernote yet installed on my new Note 3 (still contemplating the value of Evernote for me). The following worked for me:
1. Not wanting to waste time troubleshooting Evernote, or the Note 2's S Notes app,
2. Opened My Files and navigated to All files\sdcard0\S Note\
3. Selected all the S-Notes in the S-Notes app. In my case only have five.
4. After the Notes were selected, clicked on the Share icon from within the Notes app.
5 Selected Gmail
6. Sent an email to myself
7. On the Note 3, opened the email
8. For each of the five S-Note attachments in the e-mail, clicked on the attachment
9. When prompted, selected S Note when prompted by the "Complete action using" feature.
10. Then as TheMaskedMamba said, just accept the conversion message, and all is transferred to the Note 3's S Notes app.
If I had more than five S Notes, or the file size for the individual or combined S Notes were too large to send via e-mail; then I would have gone the cloud storage services route, or just copied the files to my home NAS, and then from there to the Note 3. Another option would be to try an NFC or S Beam based transfer between the two phones. Yet another could be to copy/paste files from the Note 2 to a desktop/laptop, and then from there to the Note 3 (this option is a bit redundant, but should work just a well).