New messages getting delievered to two phones after restoring a backup
You probably used the same Apple-ID for the new phone, and then all the iMessages get forewarded.

If you want to have a closer look, here is a good explanation of how it works:
selfsolve.apple.com

in short, you can do the following steps, to separate the messages on the two devices:


Transfer your SIM card to your old iPhone.
Go to Settings
Tap Messages
Turn iMessage off
Go back to Settings
Tap FaceTime
Turn FaceTime off