I use Apple's Notes app fairly often. It's come a long way from early days but it has its share of bugs. The two I run into most often are formatting bugs and especially sync failures.
Formatting bugs seem to be related to frequent edits to a note using Notes app styles and character formats. Sometimes editing stops working in odd ways. I have to create a new Note and select all/paste into the new Note.
Sync bugs are more common and more serious. There are two kinds of sync bugs - global application sync failure and note specific failure.
Global sync means no notes are synchronizing and perhaps no iCloud content. I've seen that with iCloud password changes. I recommend never changing an iCloud password, Apple really doesn't want it to change. I had to recently because trying to use the Stanza calendar on the Saint Paul Saints web site with Apple integration triggered an instant Apple account lockout and mandated password reset (be afraid).
If iCloud sync doesn't work after a password change and iCloud seems to be properly authenticated first try restarting your iPhone or Mac. Then, on an iPhone, try sending an iCloud email. That last may require you to YET AGAIN enter your iCloud password. With luck the credential problem may resolve for all iCloud functions.
Single note sync failure may be something I see because I'm using Notes across the latest version of iOS and Mojave. I suspect it's actually fairly common even on Monterey/iOS though.
If it's just one note that won't sync between two devices you can check out Apple's web client view of Notes. That will tell you which device is off but it doesn't help with the fix. As far as I can tell there's no sync failure resolution built into Apple Notes. If Apple's sluggish iCloud sync means you have edited one note on two devices when they weren't in full sync then that particular note will no longer sync. No error messages, no offer to sort things out, no creation of two versions, it just quietly sits in isolation.
The workaround is to create a new note and confirm that syncs. Then on each separate device move the content from the now isolated notes into the new syncing note. Then delete the notes that didn't sync on each device.
Synchronization is a problem some seem to have solved (Simplenote for example), but Apple is not among this elite group.