You should only print the output if $str1 and $str2 are both set, and then, clear them so you don't print them again.

The next is very close to your code, but kind of works. You only need to format the output as desired (I don't know where the large numbers come from):

#!/usr/bin/perl
open(FILE,"align.input") or die "can not open file: $!";
while($var=<FILE>){
    if($var=~/^sxoght:/){
        ($str1,$str2)=();
        @ar=split(/\s+/,$var);
        print ">$ar[2]\t$ar[8]\t$ar[9]\t$ar[1]\t$ar[5]\t$ar[10]\t$ar[6
+]\t$ar[3]\t$ar[4]\t$ar[11]\n";
    }
    if($var=~/^Query:/){
        $str1=$var;
        $str1=~s/^Query:\s+//g;
        $str1=~s/\d+\s+//g;
        $str1=~s/\s+//g;
    }
    if($var=~/^Sbjct:/){
        $str2=$var;
        $str2=~s/^Sbjct:\s+//g;
        $str2=~s/\d+\s+//g;
        $str2=~s/\s+//g;
    }
    if(defined $str1 and defined $str2) {
        for($i=0;$i<=length($str1);$i++)
        {
            if(substr($str1,$i,1) ne substr($str2,$i,1)){
                # this is not in the desired format, yet
                print substr($str1,$i,1);
                print substr($str2,$i,1);
                print "$i\n";
            }
        }
        ($str1,$str2)=();
    }
}
[download]

For the format, I propose to store the results in an array, and print "mismatch:" only if the array isn't empty at the end.

        my @mismatch;
        for($i=0;$i<=length($str1);$i++)
        {
            if(substr($str1,$i,1) ne substr($str2,$i,1)){
                push @mismatch, "$i." . substr($str1,$i,1) . substr($s
+tr2,$i,1);
            }
        }
        if(@mismatch) {
            print "mismatch: @mismatch\n";
        }
[download]

After that modification, the output I get for this file is

>hit    tstart    tend    #query    qstart    matches    qend    score
+    probability    mismatches
>gi|122939163|ref|NM_000165.3|    1595    1630    SNPSTER4_104_308EFAA
+XX:1:1:1694:128    1    35    36    -10    1.000000    1
mismatch: 30.GA
>gi|113412254|ref|XR_018775.1|    1578    1613    SNPSTER4_104_308EFAA
+XX:1:1:1608:94    1    36    36    0    0.090884    0
mismatch: 3.GT 34.TG
[download]

p.s. There's a possible speed improvement if you XOR (^) the two strings, you'll get a string of null bytes for where they are the same and non null where they are not:

        my $xor = $str1 ^ $str2;
        while($xor =~ /[^\0]/g) {
            my $i = pos($xor) - 1;  # or: $-[0]
            push @mismatch, "$i." . substr($str1,$i,1) . substr($str2,
+$i,1);
        }
[download]