I have a mac just updated to macOS Sierra (10.12) today and an iPhone 5s just updated to 10.2 today. However, this problem occurred before the updates too. I use Notes on both devices with my iCloud account. When I create a note on the mac, it shows up on the phone. When I edit on the mac those edits show up on the phone. However, no new notes or edits from the phone show up on the mac.
This leads to silly effects such as: if I create note "aa" on the mac, I will have "aa" on the mac and "aa" on the phone. If I edit to "aabb" on the phone, the mac version stays as "aa." The, if I edit the mac version to "aacc" the phone version becomes "aabbcc" or "aaccbb."
I tried turning off iCloud notes on the mac. I tried turning off iCloud notes on the phone. I tried turning them both off at once, and then restarting both devices while the accounts were off on both. I'm running out of ideas.