What is the difference between the command "write" and "write memory"?

Both the "write memory" command and the "copy run start" command save the running-configuration to the file labeled startup-configuration.

Each of these base commands (write and copy) also have other parameters/functions.  For example "write terminal" is the same as "show run", and "copy run local-file" can be used to copy the running-config to a file other than just the startup-configuration.  

Write by itself may have a default of one of the parameters (memory or terminal).  That should be easy enough to test which one it is doing.  I have used "copy run start" for a long time, but could just as easily use one of these other commands.  

Hope this helps.

-B