With responsive design, the content of the page changes dynamically depending on the screen size of the smartphone or tablet, and all elements look organic and are easy to perceive.
While in the case of mobile website, you have a completely separate website, usually with the “m.” before it’s domain developed exactly for mobile phones.
Nowadays mobile websites are not popular. Usually we have responsive design and when screen resolution becomes more and more “mobile” at some point it changes an interface a lot.