The stats are automatically updated when you visit the correct URL. That is, the URL must not specify a specific file name.
For instance, if your domain is example.com and AWStats/Webalizer were installed in the stats directory (under public_html), then the URL to use would be http://example.com/stats/
and not
http://example.com/stats/awstats.pl
If you are still having problems, then you can manually perform an update of the stats. You may need to do this on a particularly busy website, where the stats haven't been visited in some time.

SSH method:
You will need secure shell access enabled on the hosting package for this method.
Log in to SSH
Move into the correct directory (eg cd ~/public_html/stats)
AWStats: Run ./awstats.pl -config=example.com -update
Webalizer: Run webalizer -D dns.cache -n example.com ~/logs/*access_log
(remember to change example.com to your domain name)

Scheduled Task method:
This will automate the updating so you don't need to run anything manually, and doesn't need secure shell access.
Access your Online Control Panel Control Panel
Click the Scheduled Tasks icon under Web Tools
In The Run this command box, enter:
AWStats (Linux): /home/sites/example.com/public_html/stats/awstats.pl -config=example.com -update
Webalizer (Linux): /usr/bin/webalizer -D dns.cache -n example.com /home/sites/example.com/logs/example.com-access_log*"
AWStats (Windows): C:\Perl\bin\perl.exe D:\sites\example.com\public_html\awstats\awstats.pl -config=awstats.conf -update
Set the scheduled time to something sensible, such as every day at 4am (select 4am under Hour)
(remember to change example.com to your domain name, and ~/public_html/stats to the actual path the stats are installed at)
Currently Webalizer scheduled updating is unsupported on the windows hosting platform


Please note that if AWStats was installed before 26th October 2008, you will need to reinstall the package if you want to use the Scheduled Task method - as this has only recently been supported.