Show different size featured images (the_post_thumbnail) for 1st, 2nd and 3rd posts?

Just use the $count variable that you are incrementing:

<?php
    $count = 0;
    $some_featured_posts = new WP_Query(array('category_name' => 'apps', 'posts_per_page' => 5));
    while ($some_featured_posts->have_posts()): 
        $some_featured_posts->the_post();
        $count++;

        if( $count <= 2 )
          the_post_thumbnail( 'large' ); 

        elseif( 3 == $count )
          the_post_thumbnail( 'medium' ); 

        else
          the_post_thumbnail( 'thumbnail' ); 


    endwhile;
    wp_reset_postdata();
?>