use Algorithm::Diff qw(traverse_sequences);

my ($old, $new) = split /\n--\n/, join "", <DATA>; # get strings

$_ = [split /\s+/, $_] for ($old, $new); # turn them into arrayrefs

traverse_sequences($old, $new, {
  MATCH => sub { print "$old->[shift]\n"},
  DISCARD_A => sub { print "<font color=red>$old->[shift]</font>\n"},
  DISCARD_B => sub { print "<font color=green>$new->[shift,shift]</fon
+t>\n"},
});

__END__
We start. Let's begin. This is the old text. Then we end.
--
We start. This is the new text.  And here's some stuff I added. Then w
+e end.
[download]

We
start.
<font color=red>Let's</font>
<font color=red>begin.</font>
This
is
the
<font color=red>old</font>
<font color=green>new</font>
text.
<font color=green>And</font>
<font color=green>here's</font>
<font color=green>some</font>
<font color=green>stuff</font>
<font color=green>I</font>
<font color=green>added.</font>
Then
we
end.
[download]

We start. Let's begin. This is the old new text. And here's some stuff I added. Then we end.