Responsive template is a common template for both desktop and mobile versions. Advantage of this way is common codebase, so that any changes will appear on all devices instantly. Disadvantage is increasing of css files to describe page elements for different resolutions.
Mobile template is a special template for mobile devices (so, you have two templates: one for desktop and other for mobiles). This way allows to make mobile version as fast and responsive as possible, but you should sync both templates after changing colors, adding new modules, etc.
MobileJoomla is able work in both modes, but it doesn't include example of responsive template. It includes two examples of mobile templates: mobile_smartphone (based on Nokia's blank template) and mobile_iphone (based on alpha version of jQTouch framework). In addition, there are two very simple template: mobile_wap for ancient phones without support of HTML (WML only), and mobile_imode for some Japanese phones that use CHTML markup instead of HTML.