OK, here goes:
I have a legacy MacBook Pro, 2009, which cannot upgrade past Yosemite, and therefore TV Version 13.2.119192. Machine A.
I also have a MacBook Pro, Retina 2013, which currently runs Mojave, and therefore upgraded to the latest version, V15. Machine B.
When connecting from B - A, I receive the message: The remote TeamViewer is running an older version which is out-of-date. Therefore you cannot connect to this version any more.
Similarly from A - B.
I have since downgraded Machine B to exactly the same version as A, and now both machines show the same error (The remote TeamViewer is running an older version which is out-of-date. Therefore you cannot connect to this version any more.) when trying to connect with the other.
Please let me know which versions will work on which machine, so that I can connect them either way.
Many Thanks,
Ed