Could you explain what the advantage here is in using a constant?

Sure - it's the most obvious advantage: you won't be able to change them accidentally. If you do so it will give a compile-time error. Here is my previous example with just one extra line to demonstrate:

#!/usr/bin/env perl
use strict;
use warnings;

package MyStuff {
    use constant BODPLAN => 'Have a separate dev server';
    # ... more constants here
}

print "Bod's plan: " . MyStuff::BODPLAN . "\n";

MyStuff::BODPLAN = 'Use globals everywhere!';
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If you try to run that (or even compile it) it will error out. If you don't use constants you will not get that level of protection.