Advantages of a Custom Domain

Claim Ownership
Prevents someone else from owning your blog's name
Pro URL
Your blog will have a professional URL like yourdomain.com or yourdomain.in
Pro Email ID
You can use a professional email address like you@yourdomain.com
Branding
You know you're a serious blogger. But your readers may not get that impression with a free blog domain like yourname.blogspot.com
Portability
Having a custom domain lets you change your blogging platform without having to change your blog's URL. (shifting from blogger to wordpress, for example)

Advantages of Self-Hosted Wordpress

Theme, Themes and more Themes
Use any of the thousands of Wordpress themes and customize it until it truly represents you and your content.
Plugins
Choose from a plethora of plugins to enhance your blog. If you have a specific need, chances are - there's a plugin out there, waiting to do the job.
Monetization
You can monetize your blog with complete freedom - and keep 100% of what you earn.
FTP
You'll have complete FTP access to your blog.
Analytics
Unlike wordpress.com free hosting, you can use 3rd party analytics like Google Analytics to give you detailed stats.