After copying the files for the new version overtop of the old ones, the only other step should be to run the upgrade process after logging in, there should be no services or anything else on the server that needs restarting.
Do you get any sort of a notice after you replace the files and then login?
If you go to your system information page (Administration -> System Information), what do you see on the top line? It should look something like this, "DomainMOD Version: 4.16.0 (4.16.0)".