I had the same exact problem: admin tries to embed video, no problem. Author tries to embed same video (same exact code) and it shows up as a link instead.
to change this you have to modify the default user capability. Thankfully, this is easy to do.
Open up the functions.php file in your theme folder. At the top of this file add the following code:
// get the "author" role object
$role = get_role( 'author' );
// add "organize_gallery" to this role object
$role->add_cap( 'unfiltered_html' );