Are you sure that the entire HTTP dialog is the same? With the help of LWP::ConsoleLogger::Easy you can inspect deeply the entire LWP dialog with the server to double check.

Also be sure that Firefox web dev tools (you are looking at the net tab I guess) have the "preserve session" option checked (I can't recall the exact name at the moment) to prevent a clean up in between redirects.

Also check out that the "common" header values are the same e.g.: which UserAgent are you sending with LWP ?