The standard metric for that is called the Levenshtein distance. A quick CPAN search suggests Text::Levenshtein or Text::Fuzzy. The latter also has the ability to show you what edits were made in which positions.

Note that the Levenshtein distance allows for substitutions in addition to insertions and deletions. I'm not aware of a "standard" edit distance metric that only uses insertions and deletions, but you could fiddle with the weights in Text::WagnerFischer to rule out substitutions.

#!/usr/bin/perl

use feature qw/say/;
use warnings;
use strict;
use Text::Levenshtein;
use Text::Fuzzy;
use Text::WagnerFischer;

my $old = "This is an old question.";
my $new = "This is a new question.";

say Text::Levenshtein::distance($old, $new);

my $tf = Text::Fuzzy->new($old);
say $tf->distance($new);

my ($distance, $edits) = Text::Fuzzy::distance_edits($old, $new);
say $edits;

say Text::WagnerFischer::distance([0, 1, 999], $old, $new);
[download]

This prints:

$ perl test.pl
4
4
kkkkkkkkkdkrrrkkkkkkkkkk
7
$
[download]

Take your pick.

One thing you might do is to google for the Baeza-Yates k-mismatches algorithm, the Wu-Manber k-differences algorithm and the longest common subsequence (LCS) algorithms.

Some additional modules of possible interest: Jarkko Hietaniemi's String::Approx and Mark-Jason Dominus' Algorithm::Diff.