Editing footer for one page, keeping it the same for others

You’re looking for the is_home() and is_front_page() conditional tags

USe it like:

        <?php  if(is_home() || is_front_page()) : ?>
                <!-- Small footer on the home page, only has the #footer-navigation div -->
            <div id="footer-top">

                <ul id="footer-navigation">
                    <?php wp_nav_menu( array( 'container' => false, 'theme_location' => 'footer-menu' ) ); ?>
        <?php  else : ?>
                <!-- Footer for the other pages also has the #back-top div -->
            <div id="footer-top">

            <ul id="footer-navigation">
                <?php wp_nav_menu( array( 'container' => false, 'theme_location' => 'footer-menu' ) ); ?>
            </ul>

            <div id="back-top">
                <a href="#">Back to top</a>
            </div>

        </div>
        <?php  endif; ?>
        ?>