Let me see if I understand?
1. So basically copying your Sims 4 folder from:
C:\Users\<Username>\Documents\Electronic Arts\The Sims 4
2. Copy this folder to backup media of choice.
3. Then copy it from the backup to the same folder on your new computer and let it overwrite:
C:\Users\<Username>\Documents\Electronic Arts\The Sims 4
This did not work?
Or did it get copied to a different folder say for example:
C:\Program Files (x86)\Origin Games\The Sims 4
If this is what happened, The game will need to be reinstalled.
Do you still have a copy of the Sims 4 folder on your old computer. If so, copy it using the steps above.
The post linked below talks about where to find your game install files as well as where to find your user files. They are both installed in different areas. If the user files get mixed up with the install files. This will cause problems and the game will need to be reinstalled.
My Sims User Folder or Install Files are Missing?
http://www.carls-sims-4-guide.com/forum/index.php?topic=21899.msg431335#msg431335