Why is my subdomain returning a 403 status?

There are some questions I have but cannot add a comment so will do my best to help clarify the steps to set up a subdomain network: http://codex.wordpress.org/Create_A_Network.

First, the multisite feature of WordPress needs to be enabled. Did you do the steps required to enable multisite?

Second, the same wordpress files are used for the main site and any network enabled site. In other words, if you have public_html/wordpress for the domain.com then in cpanel you also point the subdomain to public_html/wordpress for a sub.domain.com. You do not create a directory public_html/wordpress/sub. The directory does not exist but the multisite feature enabled will recognize to render the WP files when sub.domain.com is requested in the browser.