The main difference between WordPress.com and WordPress.org is that one of them is a hosting site and one is not.
I have had many clients start with using WordPress.com, and the first thing I do is try to get them out of that arrangement.
WordPress.com is primarily a hosting site that uses WordPress as the platform, but it severely limits what you can do on the platform, although it does take care of everything for you (hosting, domain name, or install WordPress). There are other sites similar to this, such as Wix.com, which offer this type of service.
WordPress.com can be free if you are okay with using .wordpress as a part of your domain name (like www.examplesite.wordpress.com), but for a fee you get access to more features and your own domain name (although still limited) which most people end up doing.
WordPress.com does not allow plugins, there is no access to your backend code base, and you cannot upload your own themes.
WordPress.org is ALWAYS FREE. But using it is not.
Downloading the platform from WordPress.org is always free, but you do need to pay for certain accommodations in order to use it: a domain name and web hosting. You can learn more about web hosting and domain names here.
Many hosting sites will auto-install the WordPress platform onto your site for you, like Dreamhost.com, but you’ll still need to understand how to use the platform once you’ve installed it, so that’s where I come in! I’ll build your new site on the platform and then show you the ropes on how to manage it properly afterwards.