Hi activeleisure,
This is a good question and I am glad you asked it. For the vast majority of bloggers having "wordpress" or "blogspot" or "whatever" in their URLs is fine. If you're considering blogging seriously (as opposed to a leisurely activity) then yes your own domain is the route to take.
WordPress.com provides free blog hosting with limitations. It comes with many benefits such as (the aforementioned free hosting) automatic backups, maintenance (think upgrades, security, spam etc) the benefit of a network of hundreds of servers. Of course with free hosting comes some limitations such as the inability to install custom themes or plugins, the inability to modify the HTML, the inability to modify the PHP (core as well as theme & plugin PHP) and the inability to run ads.
WordPress.org is the home of the open-source WordPress software. The software you download for free can be used with the hosting package of your choice. This version of WordPress is restriction-free meaning the sky is the limit with what you can do with it. You can install any theme or plugin you want - heck, you can write your own. You can modify the CSS, HTML and PHP. You can modify the core PHP - the merits of which is debatable, it is probably better to create your own custom filters and hooks rather than modify core which is a whole 'nother story.

You can run ads if that is what floats your boat. Of course you are responsible for your own security and maintenance.