Free Domain Name
Free domain name is actually a sub domain of free domain name provider.
You can create as many sub domains as available on particular domain name.
Free domain name can be lost at any time for various reasons. In such case, all your Search Engine Optimization (SEO) and Marketing efforts will go waste.
WordPress.com started forced advertising or blogger banned your blog for terms of service violation.
Free domain name holders have lesser opportunities to monetize their website/blog or product.
Paid Domain Name
Paid domain name is top-level domain name. you can create as many sub domains as you can up to the limit imposed by webhosting provider.
Creating subdomain names are free.
Paid domain is under your control. However, failure to re-subscribe before expiry can lead to loss of domain name.
Paid domain is not 100% secure. They are also subject to theft and hack. So you should be alert of giving any information to third party and even domain name registrar without confirming the integrity of source from which information or action is demanded.
Paid domain name brings professional appearance to your website/blog.
A person likes to buy vegetables from supermarket may not buy vegetables from a street hawker, even he/she is offering the same at lower price.