Here are the biggest technical differences:
Blogs on WordPress.org, are self-hosted, meaning that you purchase server space through a web host. You can read more about web hosts here. The WordPress platform software is free for self-hosted, WordPress.org users.
Conversely, blogs on WordPress.com are fully hosted by the parent company of WordPress.com, Automattic. While you don’t have to worry about hosting, WordPress.com themes cannot be customized with CSS without purchasing an upgrade, and these themes cannot use plugins of any type. Please note we do not sell any themes for WordPress.com users.
If you’re on WordPress now, but not sure if it’s .com or .org, the easiest way to tell is whether you are paying for server hosting every year (note: this is NOT the same as a domain name!). If you are paying for hosting, you’re on .org.
Now for the differences in laymen’s terms:
Simply put? Control. Blogging on WordPress.org gives you full control over how your site looks and feels. Whether you work with a designer on a custom design, or purchase one of our themes, neither could be implemented on a WordPress.com site. WordPress.org gives you total flexibility and control to customize your site to your exact specifications; it also gives shops like ours the ability to bring you beautiful, easy to use themes that are built with style and lifestyle bloggers in mind.
Important for many bloggers is the opportunity to monetize their sites. WordPress.com has limitations on how you can use ad code and affiliate links on sites hosted by them, making WordPress.org the clear choice for many who are trying to turn their blog into a business. Read more about this here.
Here is more information on the differences between WordPress.org and WordPress.com