You can do this by using conditionals for mobile ie: wp_is_mobile()
Define both the sizes like
add_image_size( 'featured-cat', 248, 110, true ); //featured-cat
add_image_size( 'featured-cat-mobile', 200, 80, true ); //featured-cat mobile
and edit the code like the below:–
<?php
echo '<div class="featured-thumbnail">';
if( wp_is_mobile() )
the_post_thumbnail('featured-cat-mobile',array('title' => ''));
else
the_post_thumbnail('featured-cat',array('title' => ''));
echo '</div>';
?>