Assumption: there is only one 'mapping' in each string.

Remove the leading characters that match, remove the trailing characters that match. What's left is the difference. If my assumption is wrong, then this won't do it for you.

use strict;
use warnings;

binmode(DATA, ":encoding(UTF-8)");

my %normalize;

while (<DATA>) {
    my ($string, $normalized) = split;

    # convert to arrays (avoid unicode issues?)
    my @string = $string =~ m/\X/g;
    my @normalized = $normalized =~ m/\X/g;

    # skip matching the beginning chars
    while (@string and @normalized
        and $string[0] eq $normalized[0]) {
        shift @string;
        shift @normalized;
    }

    # skip matching end chars
    while (@string and @normalized
        and $string[-1] eq $normalized[-1]) {
        pop @string;
        pop @normalized;
    }
    my $key = join("", @string);
    $normalize{$key} = join("", @normalized);

    print "'$key'  => '$normalize{$key}'\n";
}

__DATA__
ABCÅD  ABCD
ABCÄD  ABCëëD
ABCááD ABCèD
[download]

Produces the results:

'Å'  => ''
'Ä'  => 'ëë'
'áá'  => 'è'
[download]