There are some restrictions here at wordpress.com and the basic differences are discussed in this support document.
If you decide to move over to wordpress.COM hosting, you can get the domain mapping upgrade if you wish and actually have your site here appear under your own domain name.
Having the wordpress installation on your existing web host would give you more options and flexibility as far as themes, plugins and such, but having a site here takes all the webmastery type duties off your hands. You do not have to worry about upgrades or backups and if something goes wrong, the support staff here takes care of that for you as well. There is also these great support forums with some of the best volunteers there are to help you out with things.