Mobile websites are separate sites to your main website. They are optimised with easy to read and take action on, like large tap-to-call buttons, on small screens such as smartphones.
Responsive websites are main websites which are coded to readjust when viewed on a smartphone. For some business a responsive website is all they need. Other businesses, such as restaurants, would be better served with a separate, dedicated mobile site to cater better for items such as menus etc