If you have a working Curl command, you can convert it to the appropriate LWP::UserAgent or HTTP::Tiny code using my Curl to Perl translator.

My suspicion is that the difference between Perl and Curl is that Curl picks up a HTTP proxy setting from the environment but Perl does not, or the other way around. Inspect your environment for the environment variables HTTP_PROXY and HTTPS_PROXY. Alternatively, talk to your system administrator what settings are necessary to connect to the outside world.