If you're website is fully responsive, then a subdomain with a CNAME such as "m.mywebsite.com" is not necessary. The use of a subdomain like that is for the sake of hosting a website that is specifically built for mobile phones but has the same content as the desktop website. A script is then used to detect mobile devices and redirect them from the main site at "www.mywebsite.com" over to "m.mywebsite.com".

In web development today, things have been shifting away from the old solution of separate websites since responsive design allows for easier management of the site, much better and easier to manage SEO, as well as better adaptivity to other screen sizes beyond a smart phone and desktop.

On top of that, responsive design has shifted to "mobile-first" design, which means you design for the smallest screen size, then through media queries and such, move up and up and up until you get to the biggest screen size you wish to support. This is just another nail in the coffin of the old "mobile websites".