If you pack a string into a variable and then unpack it, it must be equal, and that has nothing to do with writing to a file. Try this:

$g = "0110110001101101";
$bob = pack("B*", $g);
print $g . "\n";
print unpack("B*", $bob);
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You should get the same results. Now, try a file (remove the -l option of perl, as it inserts a new line):

open(FH, ">fh");
$g = "0110110001101101";
$bob = pack("B*", $g);
print FH $bob;
close(FH);

open(FH, "fh");
read(FH, $from_file, 2);

print $g . "\n";
print unpack("B*", $from_file), "\n";
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And report the results (note, I changed the bytes to be visible, now it should print "lm")