$sql = "UPDATE spider SET
file = '$file',
first_words = '$first_words',
path = '$path',
update = NOW($upddate)
WHERE spider_id=$spider_id";
//looks like you need this line:
mysql_query($sql); //<<<RIGHT HERE
$sql = "UPDATE sites SET
site_url = '$site_url',
update = NOW($upddate)
WHERE site_id=$site_id";
mysql_query($sql);
Also I don't know what NOW($update) is supposed to do:
Proper use of NOW() is like that -- no parameter in the parens.
I sincerely doubt that "update" column was updated by the first query, which was never executed. Maybe also you update it later or earlier in your code and just forgot you did so.