I do a fair bit of training WordPress users, be it in classroom situations, one-on-one with clients, or in meetups. I find running these sessions rewarding and challenging and I really do love seeing people leave at the end of the day or the session with that "I think I finally get it" look on their face.
In all that training one of the conversations that comes up consistently, centres around the difference between WordPress.com and WordPress.org. Specifically, the difference between a free, hosted WordPress service and downloading a free copy of WordPress and hosting it yourself.
While at its core the software is the same and the look and feel of everything you do in the dashboard to manage your site will be the same, there are some critical differences you should be aware of if you're in the process of deciding which of these options is appropriate for you.
At its most basic the difference between the two is that WordPress.com is putting your site into a network of sites all running from, essentially, one WordPress instance whereas WordPress.org is taking your very own copy of WordPress and running it for yourself.
On the face of it WordPress.com is really attractive, not the least of reasons being the fact that it's free. It's also really attractive because it's fully managed, you don't have to worry about things like behind-the-scenes security (of course, if your password is password1234 then you're on your own), updates and maintenance; WordPress.com take care of those things for you. Finally, the dashboard on WordPress.com is secured behind SSL, another useful security measure.
There are things that you have to trade for the free factor though, I'm afraid. Free WordPress.com is ad supported, so you'll have to live with ads unless you pay to have them removed. Your server space is limited to 3gb which isn't too bad if you're not using heaps of photos (or you're posting your photos via Flickr or some external site), but if you did want to add things like audio files you'll have to host them elsewhere, or upgrade your account as you can't host audio files on WordPress.com without a space upgrade.
If you want your own personal domain - such as mycorneroftheweb.com - pointing people to your WordPress.com site you'll have to pay for domain mapping. Truthfully, it isn't expensive at $13.00USD per year, and if you buy that domain name from WordPress.com then that will cost you $5.00 per year, as well.
That said, there are a lot of additional features you can add to WordPress.com to make it better/smarter/more feature-rich such as space upgrades, VideoPress, custom CSS, ad removal, premium themes, and so on. Getting across what those features are, and their associated costs may help you decide. You can get more details about that here.
The biggest drawback of WordPress.com is that that you can't extend your site with plugins or much of the cooler functionality available for self-hosted sites.
You are also restricted from serving your own ads like Google AdWords or other ads from affiliate type services. It may be considered a downfall that you have to pay to upgrade features on WordPress.com whereas with self-managed, one-off annual hosting fees should cover you altogether (except for where the add-ons you're wanting to include may be premium ones).
This may be an issue if you're wanting to paste something like a funky Twitter widget, or sign up form from some other provider. Basically, with WordPress.com you're locked into using only those things that they deem safe for the whole network (and fair enough, they've a lot of people to look out for).
WordPress.org - Self Hosted
If you want full control, self-hosted WordPress is the way to go.
When you set up an installation of self-hosted WordPress, you gain a whole measure of flexibility that you don't have with WordPress.com. There is a huge library of plugins and themes - both free and premium - available with which you can extend and customize your site. You can do whatever you want with your WordPress install!
The con with full control is that it comes with full responsibility for maintenance and for security. Keeping things like plugins, WordPress core, and themes up-to-date will be up to you or your web developer.
You are allowed to customize your theme as much as you like without any additional costs to add that functionality, you can install plugins for e-commerce, membership, and any number of other things you'd like to make WordPress do.
However, if the plugins you use introduce security vulnerabilities it will also be up to you to fix up and clean up after any breaches that may occur. Finally, you'll also have to be responsible for keeping your site backed up. Whether that's using a plugin, or using the tools that your host offers (check this with them so you know what their backup plan for you is).
To make a self-hosted WordPress happen you'll have to buy a domain name and web hosting yourself as well as set things up yourself. There are articles and tutorials out there to make that easier, but do your research before you launch in. Cheap hosting comes at a cost, so find reputable hosts with good reviews and remember, you get what you pay for. More than a few people I know have paid for cheap hosting and needed to move from it in very short order because their hosting didn't keep up with the demands of their site.
So while with self-hosted WordPress you have a whole lot more of the world at your fingertips, you're introducing a level of complexity and responsibility to the process. For some, like me, that's the fun bit; if it sounds like a nightmare, well, again, maybe WordPress.com is your best bet.
The biggest pros for WordPress.com are that it's free, secure and managed for you and a great place to get started, with the flexibility to upgrade (for a price) as you need it. You can then move from .com to .org when you're comfortable getting into some of the finer, more technical aspects of website management. The biggest con is that there are restrictions on what you can and can't do.
The biggest pro for WordPress.org is the sheer extent of options you have to do whatever you wish with your site, the con being the learning curve and the responsibility that goes with that freedom.