The first question should be "are they the same, and if yes, which one is the correct one?"

$ $ perl -wE 'say "ouch" if "foo\n" =~ /^foo$/'
ouch
$ perl -wE 'say "ouch" if "foo\n" eq "foo"'
$ # no output
[download]

So they are not the same... which one do you want?