The difference between adaptive web design and responsive web design is wider and more important than distinctions that have been suggested in this thread. The difference is neither a mater of where software functionality resides nor what unit of measurement is used in CSS conditions.
Neither term (adaptive or responsive) is a brand, so we should not deviate from the basic definitions of the words. In computer science, a response is an action, event, or message generated upon some stimulus. This definition originated from biology. The dilation of pupils in response to light is a responsive design.
Adaptation is denotes much higher functionality than simply a programmed response. The ABILITY TO DEVELOP pupils that can dilate is an adaptive design. Adaptation requires the storage of history and its later application. In biology, the adaptation requires DNA to store adaptations. In web design, adaptations could be stored in cookies or the user's account profile on the server.
Let's start with the simpler of the two. A good formal definition of a responsive web design is this:
A responsive web design examines display characteristics and instantaneously respond in the display of pages in ways that go beyond the built in automatic layout capabilities of HTML to provide a convenient, functional, and maximally visible experience across varied display conditions.
Portable devices have raised the importance of this additional size responsiveness. Many such design techniques center around the addition of conditions to the CSS (cascading style sheet or sheets) or via scripting (such as JavaScript). Each condition based on display characteristics improves the overall user experience by controlling the values of specific style parameters for a set of document elements (tags), identified by CSS selectors.*
Using either the dictionary definition or the common computer science definition of ADAPTIVE, the document layout of an adaptive web design must vary intelligently based on some higher criteria than a simple static size threshold, criteria that are evaluated continuously. A good definition of an adaptive web design is this:
An adaptive web design records the patterns of usage and conditions of use and adapts, over time, to more quickly, comprehensively, or individualistically provide users with content and functionality.
Some of the other uses of the term ADAPTIVE in regard to web design gives too much credit to what is merely another responsive design methodology, no smarter or more adaptive than any other.
Simple adaptive schemes can be programmed into JavaScript to vary CSS based on JSON data returned from RestFUL calls to the server to acquire user scroll and click stats from a database via SQL or NOSQL. More advanced user experience analytics might use a rules based system (such as DRules or Prolog) or fuzzy logic, neural net, or Bayesian schemes that run asynchronously.
An example of a simple rule is, "Sort the links in order from the most frequently to the least frequently clicked and every 10 displays put a less popular link in second place to ensure elements can trickle up over time."
Trivial feedback forms are common feedback mechanisms.