As we are going to see in WordPress.com, you hardly have any possibilities or alternatives, and the moment you want to do more specific things to customize your project you will be very conditioned. That's why my advice is to use WordPress.org since you will have the possibility of doing what you want in your project.
To be more explicit about the differences between both WordPress options, we will compare elements such as price, knowledge, flexibility, design, plugins, web analytics and online store.
Price As we have commented is free, and you do not need to hire a web hosting or a domain to use it
Knowledge The level of knowledge you have to have and the learning curve is low
Flexibility You hardly have the flexibility to develop and customize your project
Design You do not have much freedom to modify the web design, as it can be with the use of themes or WordPress themes
Plugins They are not available in this version and therefore cannot be downloaded or used
Web analytics You can not implement a tool as essential in web analytics like Google Analytics. However, the CMS itself offers its statistics from its platform
Online store It is not possible to create or manage e-commerce since you can not use woocommerce
Price, The download and use of WordPress, is free, but you need to host it in a hosting and have your web domain
Knowledge The level of knowledge you have to have and its learning curve is higher
Flexibility You have total flexibility both to develop and customize your project
Design You have all the freedom to modify the web design, as it can be with the use of WordPress themes or themes
Plugins All the plugins are available, and you can download them and use them as you want
Web analytics You can implement tools such as Google Analytics or install web analytics plugins
Online store It is possible to create and manage e-commerce with the woocommerce plugin
With Wordpress.org, you host your own blog or website. WordPress.org is where you’ll find the free WordPress software that you can download and install on your own web server.
Wordpress.com, on the other hand, takes care of all of the hosting for you. You don’t have to download software, pay for hosting, or manage a web server.