How about a Benchmark where the strings do match?

#!perl -w

use strict;
use Benchmark qw/ cmpthese /;

my $x = 'The Monastery Gates';
my $y = 'ThE MoNaStErY GaTeS';

cmpthese(-(shift),
         {
          eq => sub{ if (lc $x eq lc $y) {} },
          re => sub{ if ($x =~ /^$y$/i) {} },
         }
        )

__END__

Benchmark: running eq, re, each for at least 3 CPU seconds...
        eq:  6 wallclock secs ( 3.04 usr +  0.01 sys =  3.05 CPU) @ 24
+2974.43/s
(n=741072)
        re:  8 wallclock secs ( 3.00 usr +  0.00 sys =  3.00 CPU) @ 86
+303.00/s (n=258909)
       Rate   re   eq
re  86303/s   -- -64%
eq 242974/s 182%   --
[download]

That's not all, though. When the strings don't match and $y is shorter, eq comes out about 220% ahead. When $y is longer and the regex optimizer can tell right off that $x is too short for /^$y$/ to match, eq still has a 40% advantage.