Aside from
use strict;
use warnings;
the next major point for debugging problems in web-related code is to check your web server's error log to find the error message generated when things went wrong.
In this particular case, I expect that you'll find an "access denied" error indicating that the user the web server runs as (probably "nobody", "apache", "httpd", or "www-data", depending on your OS and distribution) doesn't have permission to create or append to /var/www/cgi-bin/new.txt. The reason it works when you run it from the command line is that you have permission to create and write to that file, but the web server doesn't, so it fails when run by the web server.
If my guess as to the cause is correct, then the solution is to first create that file from the command line, then give the web server user permission to write to it. Do not give the web server user write permission to the entire /var/www/cgi-bin/ directory, as that would open up huge security holes allowing anyone who compromised one of your cgi-bin scripts to then create their own new scripts (or replace existing ones) on your machine.
Or, better, tell us what you're actually trying to achieve and we can probably find a way to do it that doesn't require allowing the web server process to write directly to files in potentially-sensitive locations.
