WordPress.org has the WordPress software that can be installed on any web host of your choosing. With a self-hosted site you can modify anything in WordPress to truly make your site your own creation. There a unlimited themes both free and commercial. You can even create you own theme from scratch. With a self-hosted site you can install any plugin you like or even create your own plugin. It’s all up to you. Do what you want.
WordPress.com is hosting and software all in one. With WordPress.com everything is much more restricted but is very secure and almost never goes down. You can only change what they let you. Themes are limited but you usually modify the them with header images, etc. to make it unique enough. You can also get premium paid themes the may have more modifications built in. There are no plugins with WordPress.com. You are restricted to the functionality in the WordPress.com which may be adequate for your purposes.
A unique domains, themes, other customizations and functionality can be purchased with a package upgrade.