What about "o" in owl and dog? And what about backward difference? This is not exactly what String::Diff do. If you only want to get characters from second string which doesn't match to the characters in the first string it may be quite simple to implement, but I wouldn't call this a difference.

Update: maybe you'll find the following (quick and dirty, sorry, I don't yet have my tea) code useful:

use strict;
use warnings;
use 5.010;

my $pair1 = ['abcdf', 'abcde'];
my $pair2 = ['Lazy dog', 'Loony owl'];

for my $ref ($pair1, $pair2) {
    say "Words: ", join ", ", @$ref;
    say "Diff: ", diff(@$ref);
    say "Diff (rev): ", diff(reverse @$ref);
}

sub diff {
    my ($str1, $str2) = @_;
    my @arr1 = split //, $str1;
    my @arr2 = split //, $str2;
    my @res = ();
    while ($_ = shift @arr2) {
        my $ch = shift @arr1 or last;
        push @res, $ch unless $ch eq $_;
    }
    push @res, @arr1;
    join "", @res;
}
__END__
Words: abcdf, abcde
Diff: f
Diff (rev): e
Words: Lazy dog, Loony owl
Diff: azy dg
Diff (rev): oony wl
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