Off the top of my head:
Hosting your own site will make it seem more professional and serious. You could also get a [email redacted] email address.
You have access to more themes, and you can access more of the code if you want to really, really customize the look of your blog.
You can open an ecommerce store, if you want to run an online business. You can make money off ads on your site, or opt to not have any ads at all.
You're not as beholden to WP terms of service. You get more storage. Your theme can't change on you without notice.
You can use more customized analytics tools to monitor your traffic.
But really, to me, the best benefit it that makes the world take you, and your voice, more seriously. Any hobbyist or amateur can get abcd.wordpress.com--which is actually one of the GREATEST thing about wordpress. But you won't seem as fly-by-night if you actually own your own domain.