They are two different options and once you understand their differences you can decide which is best for you based in on your individual business requirements.
A very simple analogy is the difference between renting or owning a house….Strange I know, but it’s a great starting point as you begin to understand how they differ. WordPress.com is similar to renting a house. you have a house but with restrictive power and control over what you can and cannot do. WordPress.org is like owning a house, Where you have full control and you can do anything that you want with it.
Does that make sense?
So, in brief, what’s the difference between WordPress.com and WordPress.org?
WordPress.com is a service that helps you build a website using the WordPress software,
WordPress.org is the software itself, which you can use to build and maintain a website on your own terms.