Mobile web site is a website that specifically built for mobile phones. (all templates, content, graphics etc)
Responsive web site is the regular web site that you view on PC/Laptops and optimized for mobile browsers. Here if the visitor uses a regular browser, the site will display normal. If the visitor is using mobile browser, the site will be viewed with different templates.
NOTE: You need some backend code on your web site that detects mobile browsers and redirect users to mobile version of regular site.
A mobile site is essentially a copy of your website, where the server does the work to deliver an optimized page thatâ€™s smaller and easier to navigate. The solution can really customize a unique experience for mobile users.
In responsive design, the device does the work and automatically adjusts according to a deviceâ€™s screen size (large or small) and orientation (landscape or portrait). It switches between these options on-the-fly.