On your iPhone, go to Settings > Apps > Mail > Account, and choose the email account that is not syncing. If it shows POP Account Information on the top Header, then that account will not sync with other devices. Only an IMAP account is able to sync with other devices.
You would need to delete the account and add it again. Whether an account defaults to POP or IMAP depends on the server address, so if yours is defaulting to POP, then when you enter your email address, intentionally use an incorrect password to go to manual entry. On the top of the screen, you will see the choice of POP or IMAP and you would want to select IMAP. You would then need to enter the correct Password for that account in the Password field of the Incoming Mail Server and also the Outgoing Mail Server listed as SMTP, along with the correct server address, Port Number, etc that the mail provider uses. If you don't know what those are, we can help to determine the settings to use if you let us know the same of the email provider.
With iCloud accounts you don't have to do that, but certainly with many third party mail accounts you would need to do that. There are just a few third party email providers that will default to POP that you have to manually set them up where it is not done with simply the email address/password.
For me when I set up my new phone, I used the Quick Start method where data is transferred with the old device sitting next to the new phone and did not have to make any changes to my Mail Account. Everything transferred over without any problems. Not sure if there is a difference when setting up from an iCloud Backup alone.