The exclusive-or operator (^) between strings returns \x00 for each matching pair of characters, and a different value for non-matching characters. Thus, 'Perl' ^ 'Perl' would return '\x00\x00\x00\x00'. Matching a returned string for [^\x00] will show where the strings differ. In your case, only the first difference is requested. Given this, consider the following that uses your data:

use warnings;
use strict;

open my $fh1, '<', 'first_file.txt'  or die $!;
open my $fh2, '<', 'second_file.txt' or die $!;

while ( my $s1 = <$fh1> ) {
    chomp $s1;
    chomp( my $s2 = <$fh2> );

    ( $s1 ^ $s2 ) =~ /[^\x00]/;
    substr( $s2, $-[0], 0 ) = '+' if defined $-[0];
    print $s2, "\n";

}

close $fh2;
close $fh1;
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Output:

amaya+mAn
vismaya+mAn
soura+mA
kamal+An
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The variable $-[0] contains the position of the last match, which is passed to substr to insert a + at the location of the first difference between the two strings.