Responsive website designing is an approach aimed at designing sites to present a responsive (optimal) viewing experience, easy reading, and navigation with a minimum of resizing, panning and scrolling across a wide range of devices ranging from phones, tablets to the desktop computer.

Mobile Responsive Design gives Better User Experience. Going back to SEO  for moment.  If you have a Separate desktop and mobile sites. You are going to have to run Separate SEO Campaigns and one website. That would make not just your Customers happy.