Wordpress.com
Is the "more commercial" product of them. The product that is easy to get, configure and launch. They take care of the setup of wordpress site, and you have somehow limited control. They also win from a few adds places on your website and so on. You have control over the activated theme, plugins and so on, but limited control over the platform/hosting itself.
On the other hand,
wordpress.org
Is the full control version of the platform. If you have a few developer skills, you can download the platform for free, and then buy a domain, a hosting plan, and install the platform there. You can create your own theme from scratch, or use onone of the predefined themes that comes with the platform package download.
TL;DR
If you want full control over the platform, buy your own domain, hosting, and usehead over wordpress.org for a free copy of the platform.
If you don't have developer skills and you want an easy to maintain setup, choose wordpress.com