Both responsive and adaptive designs attempt to optimize the user experience across different devices, adjusting for different viewport sizes, resolutions, usage contexts, control mechanisms, and so on. Responsive design works on the principle of flexibility where a single fluid webpage can look good on any device. Responsive websites use media queries, flexible grids, and responsive images to create a user experience that flexes and changes based on a multitude of factors.
Like the responsive design, adaptive is rooted in trying to create a better user experience across a number of different devices, but it is more like the modern definition of progressive enhancement. Instead of one flexible design, adaptive design detects the device and other features and then provides the appropriate feature and layout based on a predefined set of viewport sizes and other characteristics.
Responsive design is client-side, meaning the whole page is delivered to the device browser (the client), and the browser then changes how the page appears in relation to the dimensions of the browser window, while adaptive design is server-side, meaning before the page is even delivered, the server (where the site is hosted) detects the attributes of the device, and loads a version of the site that is optimized for its dimensions and native features. The amount of information downloaded on the user end is different, making adaptive websites a little quicker to load. In the build itself, the adaptive design uses multiple templates for a single website whereas responsive design uses a flexible framework with a single template. Some designers argue that adaptive sites are easier to build because they can be based on an existing site.
You may consider using adaptive design if a device-specific experience is a necessity. Adaptive design is specifically for mobile or tablet experience. You can actually create different experiences for each of these devices and can handle and maintain adaptive templates and resources, and if your user base is accessing your information on a lot of different devices (if analytics show 70 percent of users are on a single device, an adaptive UI might not be worth your time).