Same website but different logo for large screen and small device?
There are multiple ways to do this through CSS media queries. See example below. 

.classname {
   @media all and (min-width: 768px) {
     background-image:url('image2.png');
   }
   @media all and (min-width: 320px) {
     background-image:url('image1.png');
   }
}


Helpful links


Responsive Images
Working with breakpoints


If you use the module and breakpoints from your theme you can probably just wrap the logo in a div class so you can control the padding or other CSS attributes. Just make sure to find the correct image styles per breakpoints and multipliers.