This is happening to me to. If I send a message from the website "Verizon Messages" it does not appear on my phone. If I send a message from my phone, it does appear on the website. It just seems that messages from the website are not being pushed back to my phone.
What make/model phone do you have?
I just tested with my Droid Turbo 2 and messages sent from the website showed/synchronized on my phones' Message+ app, as well as on my integrated/connected iPad.
I have an Gallaxy S5. I was using the Google Messages app, and it wasn't syncing. So I switched to using the Message+ app and it still wasn't syncing with that app either.
roemhildtg, syncing messages should be par for the course when it comes to Message + (formerly called Verizon Messages). When did you notice that they weren't syncing? What version of the app are you running on your S5?
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