Please allow me to rewrite your examples in a way that maintains rough equivalence whilst hopefully making the differences more apparent.

{ my $a = "MIN.(NERR_VOL2)";
  my $b = $a;
  print "matches" if $a =~ m/$b/;
}
[download]

{ my $a = "MIN.(NERR_VOL2)";
  my $b = $a;
  print "equals" if $a eq $b;
}
[download]