Assuming that the WP version are the same then then only thing left is some difference in the hosting environment. I am not aware of anything that should stop the oembed field from working.
It could also be a javascript error. Are you seeing any errors reported in console? And easy way to test scripts is to add define( 'SCRIPT_DEBUG', true ); to your functions php file and see if that clears up the issue. If it does then you likely have a corrupted minified script file, this can happen when using FTP to transfer large minified script files.
