They are not identical. STDERR is a handle, automatically opened, used to display error messages. I don't find anywhere that STERR is a Perl variable (or an OS variable, either).

The main difference between the two is that STDERR is automatically opened when you start Perl (along with STDIN and STDOUT). You can print to or read from these three handles right away, without having to open them. If you want to use STERR as a handle, you need first to open it.