I found a fix
I was having major issues with the Notes app in the new Mountain Lion update. Here are the problems I was having:
Notes was displaying two iCloud accounts when I only have one.
Notes on Mac were not pushing to iOS devices.
Notes on Mac were duplicating over and over.
Tip: I would back up any notes that are important to you before attempting this. I didn't lose anything but you never know.
First, close the Notes app and open the Mail app. We are going to fix the duplicate iCloud accounts. You would think going to System Preferences would fix this but it's actually in the Mail app. I went to Mail > Preferences > Accounts. In this list there were two iCloud accounts which were actually the same account. I kept the iCloud account with my email address as the description and deleted the account the said "iCloud" in description.
After that you can close mail preferences. Then along the top settings in Mail click "Mailbox" and then "Rebuild". This re-syncs your email accounts.
Close the Mail app.
Now we are going to fix the duplicating notes issue. I found this solution on another thread(
https://discussions.apple.com/thread/4137558?start=0&tstart=0). This person actually called Apple and was told to follow these steps:
1. Click on mac HD/ library /caches. drag all the contents in the folder to the trash
2. Click on the finder click go, hold the option key and click on libraries (will not appear if you don't hold the option key). go to the caches folder again and again drag everything in the folder to the trash.
3. Click on the finder click go, hold the option key and click on containers. Drag com.apple.notes to the trash
4. Restart your computer.
5. Empty trash.
6. Try to open notes again.
And that's it. I had a few duplicates remaining, but I just deleted them and they have not come back. Things seem to be working good now. Everything is syncing between my Mac and iOS devices, including folders. The only issue is that notes with images are not syncing at all. This may be a limitation that is fixed with iOS 6 though.
Anyways, hope everyone finds this helpful