Mobile and responsive websites though essentially serve the same purpose, but they differ hugely from each other. A mobile website is essentially a reset of an existing desktop website, which is customized and designed separately to provide users with unique smartphone browsing experience.
Responsive website, on the other hand, is built with a responsive code and is the same desktop website but completely fluid in order to provide users with optimal viewing experience across multiple devices ranging from desktop to the smallest smartphones.
However, a responsive framework is given preference in front of mobile devices for various reasons. Consider these advantages:
One website for every device
Since the responsive website fits to any screen size, it eliminates the need of a separate website for mobile users. A single website with single URL and set of codes would be enough making it easier to manage the website, which is not seen with websites designed only for mobile devices with different URL.
Faster browsing experience
When designed carefully, a responsive website loads faster than a mobile website. Since they have to account for desktops as well as limited memory devices, they are designed with fewer components to offer a superbly fast browsing experience.
Since it is recommended configuration from Google, it is given preference by the search engine. With a single URL, which is not possible with the mobile website, it makes user interaction easier and helps your website gather more traffic.
Though a mobile website would offer you more user friendly attributes, a responsive website would prove more beneficial for you than just being a user-friendly website. It minimizes maintenance in the long run while also providing a more adjustable feature to any mobile user.