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.
This is because wordpress has the Author user role defined as someone who can only write and edit their own posts AND cannot post javascript or complicated html.
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' );
Basically, you are just giving the author user role access to post any kind of code they want by adding the ‘unfiltered_html‘ capability. Just make sure that you can trust all of your authors before adding this, because without html filtering, a user could do considerable harm to your site if they decided to post some nasty javascript.