Some CSS and JS is not loading for anonymous users
First of all you should find where the CSS or JS is being added, this being Drupal, there are tons of ways that this could happen.

if they're added using drupal_add_js and drupal_add_css (which they should be) you have to make sure that possible template differences beetween roles and permissions both print out the $styles and $scripts variables.

If the theme follows D7 standards there should be an html.tpl.php file somewhere (most likely the root) in the theme, that would be the best place to start looking. If that does not exist, try page.tpl.php.

Post your findings and it'll be easier for us to help you.