They are basically the same functionality as long as you are within the same Qualtrics account. If you are trying to share the survey with a separate instance / both users are not on the same account (as when your organization has different instances in place), then depending on your sharing rights, 

Both will copy the content & flow, as well as any of the reports built in the “old” reporting module. Reports built in the “new” reporting modules, custom defined variables (bucketing etc) will not carry over either way.

“Copy” means you retain ownership of the survey and need to “collaborate” it to allow another user to access it or use admin access to transfer it over. Sharing the QSF means the user who imports the QSF will own the resulting new survey.