Responsive web design is widely regarded as a must when designing for multiple devices. With just one code base, instead of multiple sites, you can more efficiently make use of your resources. But, how your design looks is only a piece of the overall experience for a user. Having it be able to adapt to different browsers and technology can fully round out the interaction.
Aaron Gustafson describes adaptive design as layering on the experience. He likens it to a peanut M&M. At the core is a peanut, which is a perfectly acceptable snack. But after layering on chocolate and then a candy shell, it arguably becomes a much more enjoyable experience. Just as on the web, if you have a more capable browser that can support the latest in CSS and HTML. you’ll get a richer experience. But even at its core, your site should work on more stripped down devices.