This is not an "Apple thing", these are fundamental concepts that are common to any platform. Ignore iCloud and iCloud Drive for a moment, they're irrelevant to the discussion.
Sync means to make a mirror image of some data source. You make a change on device "A" and the change is replicated wherever the master source of the data resides, and vice-versa. If you have more than one device, the change you made is also replicated on device "B", etc. Syncing is dynamic and responds immediately to changes you might make. If you change the data there is no recovery, so it is not the same as a "backup".
Backup means to make an archive of some set of data. It is a static record that existed at the point in time when you made the backup. If you change the live data you do not change the backup, unless you make another backup after making the change. So a backup is what you would try to recover for your new iPhone when the original was churned up by a snow plow.
So what you want is a backup, and your device can be automatically backed up to iCloud or to your computer (your choice).
But not everything is included in a backup, namely things that you might sync from some master source. These things are not part of your device backup since the information resides elsewhere -- e.g. if you lose your iPhone you don't lose your email because it resides on a remote server, so there's no need to back up your email as part of your iPhone's backup. Likewise with contacts, if your contacts are stored in your Gmail account, all you need to do is set up your Gmail account on your new device and voila, you have your contacts back. Likewise with apps, they exist in the iTunes store and can be re-downloaded for free, so there's no need to back them up (but any data they accumulate is part of the backup).
Does that help?
So your fundamental question you started with implies that Notes you enter in your iPhone are not showing up either in iCloud and/or your wife's iPhone. If that's the case, then perhaps you're not syncing Notes to iCloud ... which is entirely possible, since some people don't want their notes in the cloud and prefer to keep them only on the device (in which case, the notes are part of the backup since there's no other source).
Maybe go to Settings> iCloud and see if the Notes option is turned ON. If it is, and you still don't see your Notes on your other devices or in iCloud.com, then open the Notes app and see if there is a yellow "<" in the upper left corner. If there is, tap that and see if your notes are in the iCloud category or (new for iOS 9) the "On My iPhone" category (which keeps the notes out of iCloud).