The growth in mobile search has caused Google to change their search algorithm. A few months ago, Google announced that they now favor mobile friendly websites when a user is searching from his or her mobile phone. Essentially, Google search results are different on mobile than desktop, with mobile search rankings taking into consideration the user experience of the website on a smartphone.
However, Google was very specific as to what the best practices are in mobile friendly websites. Google did not endorse the traditional mobile website as the optimum choice but a responsive design. That goes against the conventional wisdom, at least from several years ago, and warrants deeper explanation.

What is a responsive website?

While a traditional mobile website is an alternative website formatted for mobile devices that you are automatically redirected to on a smartphone, a responsive design is one website with one code. A responsive design changes shape to fit the screen dimensions of the device you are using, whether a modest smartphone, a galactic Samsung Galaxy or a tablet. A true responsive design will actually change shape when you narrow your browser window on a desktop. That is the easiest way to tell the difference.
There are also responsive-like designs. Instead of moving fluidly by screen dimensions, these sites change into two (desktop and mobile)  or three shapes (desktop, tablet and mobile) based on sensing the approximate screen widths.  The site senses the width and switches over to different formatting based on the screen width.

Is the user experience of responsive websites inferior to traditional mobile websites?

The short answer is no. But of course, the website designer has to plan out how the shape will adjust to size widths before they build the website (as there are always transitions). The point is to avoid having to zoom in or force annoying scrolling (the problem with regular websites on mobile), meaning the text will stay the same size and the content will have to be transferred to a more vertical setup in mobile and tablets.
Like mobile websites, the navigation bars and toolbars are condensed, normally becoming a simple menu for easy usage on smaller screens.

Why does Google like responsive website over separate mobile websites?

We can only guess, but a true responsive website  should take care of the seemingly innumerable screen widths and compatibility issues.  Additionally, Google is very concerned with the authority, trustworthiness and similarity of a website. Since there is a redirect with a mobile website, there can be different experiences, including different content (with often a lot less content through a smartphone). Also, mobile redirects create the potential for security issues that Google needs to take into consideration with ranking. So Google wants the same HTML code on one page, which is exactly what a responsive design is.

Will having traditional mobile websites have a harmful effect on search?

The quantification of this is hard. But according to Google’s own best stated practices, a responsive design (one HTML) is the best, followed by a traditional mobile design (of course, emulating the content on the original website). The worst option is only having a desktop version, making it very hard to read and navigate on mobile.
Responsive design websites are the future as equipment makers do not make standardized devices and browsers have a lot subtle differences. They provide better experiences and also address how Google’s and other search engines rank websites. In our opinion, if you want to do well (or keep doing well) in search, it is time to make the switch over to a responsive design.