Mobile website :
Mobile website is a separate site, designed for mobile phones and mobile browsers.
The main benefit of having a mobile site is that you can target your mobile audience very well.
The drawbacks are several. Since your desktop site and mobile site are two separate entities, when you update your website, you will need to update your mobile site as well.
Responsive website:
Responsive website is a single website that scales based on what device was used by the visitor – desktop or laptop, mobile phone, or tablet.
There are many benefits to responsive website.you only have one single site, It requires little maintenance once implemented and is a good long term solution. A responsive website benefits from exposure to all visitors – mobile or desktop – and steadily gains popularity on the website because of links and sharing.
There are still some drawbacks in it. it will take a lot of resources to design and implement the new website layout. The website content then needs to be reviewed and edited to accommodate both desktop and mobile viewers.
Some Useful Links :
https://ux.stackexchange.com/questions/13273/responsive-web-design-vs-separate-website-for-mobile
http://noldorweb.com/blog/responsive-web-design-vs-separate-mobile-website-design/
http://www.smartz.com/web-development/mobile/mobile-vs-responsive-design/