Whats the difference between copy running-config startup-config and Write Memory On a Cisco Switch or Router?

The difference is that the write mem command is being discontinued. It is being replaced by the copy command and is the one you should use on the Cisco tests. I still use this command on some routers such as 831, 871, 3002, 2950, 2600 however it appears that as of iOS 12 it will be gone.

Hope this helps!