Using `auth_redirect` : keeps asking me to login even when I'm loggen in

Question

I am trying to use auth_redirect to automatically redirected not logged-in visitors when then visit a specific page. Here is the code I use:

add_action('template_redirect','wpse16975_check_if_logged_in');
function wpse16975_check_if_logged_in(){
    $pageid = 29;
    if(is_page($pageid)) auth_redirect();
}

The redirection works (I see the login page), but when I log in, I see the login page again. There is no way to access to the protected page because the browser keeps redirecting it to the login page.

I saw this question and its answer from @chrisguitarguy and modify my code to this :

add_action('template_redirect','wpse16975_check_if_logged_in');
function wpse16975_check_if_logged_in(){
    $pageid = 29;
    if(!is_user_logged_in() && is_page($pageid)) {
    $url = add_query_arg(
        'redirect_to',
        get_permalink($pageid),
        site_url('wp-login.php')
    );
    wp_redirect($url);
    exit;
}

This code solved my problem : a logged in user can now access to the page. The question is : why cannot simply use auth_redirect ?