menu in different page in different style

you can use this,

<?php if( is_front_page() ):?>
<div class="home-page-menu">
 wp_nav_menu( array( 'container_class' => 'menu-header', 'theme_location' => 'primary' ) );
</div>
<?php elseif; ?>
<div class="inside-page-menu">
 wp_nav_menu( array( 'container_class' => 'menu-header', 'theme_location' => 'primary' ) );
</div>
<?php endif; ?>

Here, you should be change your CSS style for home-page-menu and inside-page-menu .
for more details about this is_front_page() follow this link
http://codex.wordpress.org/Function_Reference/is_front_page

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