I'm guessing that you installed the code into the primary domain's cgi area (public_html/cgi-bin/). I don't think the subdomain, which is technically a separate website, can access that. You may need to install your code into public_html/subdomain/cgi-bin/.

If you need a separate /cgi-bin directory, you'd have to manually create one via cPanel > File Manager or an FTP client. First of all, where is the HTML sending the data via the action= attribute? Is it action="/cgi-bin/counter.cgi" (or similar), or is it action="http://subdomain.primary.com/cgi-bin/counter.cgi"? The idea is that the primary domain, subdomains, and add-on domains are all seen as independent sites, and can't access each other's directories and files directly. "/" is different for each of them. You have to go through "http://domainname/path/file" addressing. Are you getting any error messages that suggest that the server is unable to access something?

When you say "subdomain", do you mean "www.adifferentdomain.com" or "subdom.primarydomain.com"? The first case is an add-on domain and the second is a subdomain, although the CGI problem should be the same in either. Just keep in mind that you can't jump back and forth between the primary domain and subdomains or add-on domains by using just a directory path. You can with http://....., but be careful.