There are several key differences between these two variations of WordPress. Most importantly, they differ in hosting solutions. WordPress.com provides its users with free and automated security upgrades, backups, etc… but offers zero control over hosting.
On the other hand, WordPress.org is a self-hosted solution which offers way more flexibility, ability to upload custom themes and plugins (.com version only has about 200+ that you can choose from) and complete control to edit the code any way you want.
However, you are responsible for backing up your own data and eliminating security threats.