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; ?>
?>