Display the number of unseen comments on a page since the user last visit

You can store the data in the user meta as an array of post-id -> comment count at last visit and then simply count the comments since that date, for example

function get_user_comment_count_since_last_visit($user_id ,$post_id){
    //only do this for logged in users
    if ($user_id <= 0 ){
        return 0;
    }

    /**
     * get last comment count for a post id from user meta if set
     * and if not set then we define zero
     */
    $user_last_visit = get_user_meta( $user_id,'_last_visit_',true);
    if (!isset($user_last_visit[$post_id]) || $user_last_visit[$post_id] < 0)
        $user_last_visit[$post_id] = 0;

    /**
     * get current comment count of the post
     */
    $comments_count = wp_count_comments( $post_id);

    /**
     * get the amount of added comment since last visit
     * and update the user meta for next visit
     */
    $user_last_visit[$post_id] = $comments_count->approved -  $user_last_visit[$post_id];
    update_user_meta( $user_id, '_last_visit_',  $user_last_visit);
    return $user_last_visit[$post_id];
}

and to use it you call it with the page id and user id of the current logged in user.