Drupal 7 with Media: YouTube only embeds static image
Media 7.x-1.x only embeds images via WYSIWYG.
I would always recommend media 7.x-2.x for anyone unless you have a strict security policy of only using full releases. 2.x has many great new features.

With Media 7.x-2.x:

The media module currently only has support for embedding images via WYSIWYG.

What version of media are you using? If you use media 7.x-2.x you can get it to work by using this sandbox module - http://drupal.org/sandbox/DevinCarlson/1823634 (note that you also need separate http://drupal.org/project/issues/file_entity module with 2.x because it has now been split out of media - it was part of media in 7.x-1.x)

That sandbox module allows configuring which view modes are used for different file types when embedding via WYSIWYG.

There is a related issue for the media module here http://drupal.org/node/1792738 - It has been decided in that issue that the sandbox module will be integrated into the media module. So you can safely use the sandbox knowing that the functionality will eventually be in the media module, at which point you won't need to rely on a sandbox project.

When using that setup of modules I find that in a lot of cases it is also good to use http://drupal.org/project/entity_view_mode - which allows you to create new view modes (the file_entity 7.x-2.x module doesn't provide many view modes out of the box).