The difference between WordPress.org and WordPress.com is who is responsible for hosting your website.
Through a third-party hosting provider, you host your own website or blog on WordPress.org. You must also install WordPress, purchase a domain name from a third-party site, and maintain your server. It's a far more interactive experience than WordPress.com.
WordPress.com provides hosting for your website. You won't have to install any software or manage a server either. Your website's URL will be www.qwertyexperts.wordpress.com if you choose WordPress.com. You can, however, upgrade your WordPress.com account and purchase a custom domain from a third-party vendor (meaning your URL will look like this: www.mywebsite.com).