How to have the Header show different content on different pages?

It doesn’t sound like you need another Navigation menu.

How about something like this:

<?php if ( is_page() && $post->post_parent > 0 ) { 
   // we know we are on a sub/child page now...
   // PS. another way to handle this would be to check if you were on a particular page template or particular page ids etc.
?>
<a class="awesome-button" href="https://wordpress.stackexchange.com/questions/332869/<?php echo esc_url( home_url("https://wordpress.stackexchange.com/" ) ); ?>">Go home!</a>
<?php 
    // just show a link Home for the subpages...
} else {
   // now we are NOT on a subpage, so show the Primary nav menu...
   wp_nav_menu( array('menu' => 'primary') );
} ?>