Adaptive Web Design offers all the benefits of Responsive Web Design and solves the lacks with RWD such as, that the user don't get a completely optimized experience for the device, searchers behave differently when using mobile devices and since all versions are served the same content websites can take long to load on a smaller screen, damaging the overall experience. In Adaptive Web Design content not only adapt to the screen size, it also takes into account the capabilities of the device as well.
This offers a much better user experience as it adopts a different design for touch screen devices and a traditional mouse & keyboard arrangement. Multimedia features are completely optimized to speed up the loading times on a page.
Responsive Web Design is the subset of Adaptive Web Design, AWD includes few other features that makes use of the capabilities of the users device where as RWD don't. The extra feature in Adaptive web design is often termed as Progressive Enhancement.
Progressive enhancement is the practice of optimizing to the lowest common denominator and selectively adding functionality based on the capabilities of the device when moving up.