Responsive web design is not mobile first. In fact, in most cases, it’s mobile last.
Responsive means taking an app or a site that is web-first and making that experience mobile-friendly.
Mobile first, on the other hand, entails thinking about usage on mobile and extrapolating the flows and user experience to be seamless on the web as well. The constraints of a mobile environment are powerful. They force us to tailor app development and present only the most important actions to a user in a streamlined way, so that an average user can navigate the app in expected ways. Mobile first is paring down redundancy, starting with strong information architecture and intuitive hierarchical menus, and building that into every form factor. Mobile first does not mean that every screen should look the same nor have 1:1 feature parity, just on a littler screen with a slightly different layout.
Take caution when you approach the land of “mobile first,” lest you get swept up in tiny versions of web apps and call it a day.