First form, WordPress.com, is a commercial service that offers free and paid blogging tools (using WordPress of course) on the web. WordPress.com uses the software as a service model (a.k.a. SaS), maintaining the blogging software tools and taking care of things like security and content delivery (bandwidth, storage, etc.).
The second form, WordPress.org, is the community that helps develop and maintain the open source version of the WordPress software. The whole of the WordPress blogging tool can be downloaded and installed to a computer, server, or hosting provider of your choice. The setup is in your hands and you are responsible for providing the necessary security and content delivery.
Why would you pick one over the other?
Let's start with why WordPress.com first. Remember, they provide the software ready to go as a blog. The setup you are responsible for, if you want, is designing the look of your blog. Things like themes or layout are available for you to organize. There are defaults and WordPress.com offers suggestions. WordPress.com also offers a good sized set of widgets and plugins, which are mini-blogging tools that add features and functionality to your blog. For example, you want an index of past blog posts? There is the Archive widget. Want to show your latest photos from Flickr? There is a Flickr widget.
WordPress.com is also a commercial business, offering extra items to help enhance your blog. These extras have a price, though not expensive at all, and help buildout your blog even more. For example, the default themes are pleasant enough to start blogging. But if you want some visuals or layout to more closely fit your style, then you may want to purchase a premium theme.
When you start a blog on WordPress.com, in the free version, you will receive a domain name that looks like this: your-blog-name.wordpress.com. For example: farmerbrownsays.wordpress.com. To have have a non-wordpress.com domain name, you will need to upgrade your service to use a custom domain name.
WordPress.com is, again, a commercial business so they may, from time to time, run ads on the free blog sites. You can avoid having those ads show up on your blog by purchasing the Value Bundle. The Value Bundle also provides for additional space (important if you have lots of pictures), allows you to have a custom theme, and a custom domain name.
There are some restrictions on WordPress.com usage that you may need to consider. Using any plugin you want is not be possible if WordPress.com does not already provide it their official service. For example, do you want to use the SexyBookmarks plugin? WordPress.com does not have SexyBookmarks as part of their core plugin service. Want to use the NextGen media management plugin? This too is not part of the core WordPress.com plugin suite.
This is not to say WordPress.com does not have sharing links (they do, see Sharing) or media management (this too they have, see Media library). The reason WordPress restricts the use of plugins is because plugins are software that must be maintained over time to ensure a functioning WordPress.com service. Allowing any plugin could cause the WordPress.com service to falter and, in the process, cause issues with your blog.
Why use WordPress.com? The biggest reason is for the cost, either free or the premium bundles, is lower than having to host and maintain your own WordPress.org site. Think about what WordPress.com is offering, in their free version: a blogging platform ready to go on a web server they manage and maintain. And for the premium bundles, costing from $99 to $299 (UPDATE 2013 03 13: $99 to $299 per year), they take on the labor, time, backups, and effort to ensure your blog is available and informing your audience. You can then just focus on blogging, finding those interesting ideas and sharing them with others.
What about WordPress.org, a self hosted WordPress? With all the above thoughts on WordPress.com, why would you want to even download and setup WordPress in your own part of the Internet?
The main reason many people do this is because more control. Plugins and widgets of your choice can be used. For example, if you are a photographer that wants to create photo galleries of your work then the NextGen media plugin is what you need. Or, if you want to heavily customize the look with base themes like Thesis or Genesis, then WordPress.org is for you.
If you want to run your own ads, self hosted WordPress is what you need. WordPress.com does not allow one to run affiliate ads or other similar campaigns (see note on Advertising).
A self hosted WordPress does offer more flexibility when it comes to setup and configuration. However, with that flexibility comes responsibility. You are responsible for hosting (for example on a service like BlueHost), blog software maintenance as needed (seed post on upgrading), and backups.
Which to pick? If you are a business just starting out blogging then I would recommend WordPress.com and focus on developing your blog as a practice. The reason for this is the value of your time: do you want to futz (fancy term for waste time) around? Your goal is to communicate with an audience, your customers, on a regular basis. The costs to getting started, even with the premium package, is low compared to your time.
And if you are not a business and just want to get to blogging, the WordPress.com free model is really easy to start up. Again, you don't have to futz, allowing you to focus on the content and practice of blogging.
After six months, or so, of blogging (weekly, right?) you might want to revisit your use of WordPress.com. Think about the business or blog critical needs that have not be met. With those unmet needs in mind you can make a decision on migrating to a self hosted blog or not. And (here's a really great feature) the migration from WordPress.com to WordPress.org is pretty straight forward. It will require planning and testing but the process is well known.