Step 1 has to be to check whether any means of establishing an HTTP connection from the bad (second) system to the destination works. Try wget/curl/lynx, etc. If none work, then it is clearly not a perl problem.

Conversely, if you can get a working HTTP connection from the bad system, it's time to examine the differences between the bad system and the good system. Same O/S? Same version of perl? Same version of LWP?