I don't know why you have a second output there, maybe I misunderstood something.

Nevertheless: here is my attempt which assumes (= doesn't verify) that both strings are identical except for the number of dashes.

#!/usr/bin/perl
use strict;
use warnings;

my $oldstring = 'ATTGC---AGTCCATGC------ATGC';
my $newstring = 'AT-TGC---AGTCCATGC--------ATGC';

my @oldstring= ($oldstring=~ /(-*)(?:A|C|G|T|$)/g);
my @newstring= ($newstring=~ /(-*)(?:A|C|G|T|$)/g);

my $pos= 0;
for (my $i= 0; $i < @oldstring; ++$i) {
    if ( $oldstring[$i] ne $newstring[$i]) {
        print length($newstring[$i]) - length($oldstring[$i])," at pos
+ition ",$pos,'-',$pos+length($oldstring[$i]),"\n";
    }
    $pos+=  length($oldstring[$i])+1;
}
[download]

Result
1 at position 2-2
2 at position 17-23

It works by collecting all groups of dashes in front of any letter or the line end. Groups may heave length 0. Then both results are compared.