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.