Here is a simplistic algorithm that will just compare each line, one at a time, and show whether the individual lines match or not.

use warnings;
use strict;
use autodie;
use File::Compare;
my $F1="version1.txt";
my $F2="count.txt";
my $F3="differ.txt";

#############################
#### adding these sections to create the files for me
{
    open my $fh, '>', $F1;
    print {$fh} <<EOT;
This is line one
This is second line
This is third line
EOT
} # automatically closes file when leaving scope

{
    open my $fh, '>', $F2;
    print {$fh} <<EOT;
This is line 1
This is secont line
This is third line
EOT
} # automatically closes file when leaving scope
#############################

my $cmp = compare($F1,$F2); # [pryrt]: if they're big, it's better to 
+compare once, rather than three times
    #the addition of the USE line is important for this function to wo
+rk.
if($cmp==0)
    {print"they are the same\n";}
elsif($cmp==1)
    {print"they are different\n";
    #opening/creating all three
     open my $fh1, '<', $F1;
     open my $fh2, '<', $F2;
     open my $fh3, '>', $F3;
     while (<$fh1>)
     {
         last if eof($fh2);
         my $content2=<$fh2>; #reads the complete file to content2.   
+  # pryrt: add 'my' for lexical scope
         chomp($_, $content2);
         #haven't yet figured out how to pull out the difference data.
         #### [pryrt]: here's a simple comparison, just highlighting w
+hich lines are differnt
         if($_ eq $content2) {
            # this line is the same
            printf $fh3 "%-20s= %s\n", 'MATCH', $_;
         } else {
            # this line is different
            printf $fh3 "%-20s> %s\n", 'DIFFERENT', '';
            printf $fh3 "%20s< %s\n", $F1, $_;
            printf $fh3 "%20s> %s\n", $F2, $content2;
         }
         #### [/pryrt]
     }
}
elsif($cmp==-1)
    {print"error\n";}
else    {print"something wrong\n";}

print"Please check the file $F3\n";        # pryrt: don't repeat yours
+elf: use the filename variable

### pryrt: I want to display F3 without external help
{
    print "\n===== $F3 =====\n";
    open my $fh, '<', $F3;
    print while(<$fh>); # uses perlish postfix, and perlish default of
+ print using $_
    print "\n===============\n";
}
[download]

with output:

they are different
Please check the file differ.txt

===== differ.txt =====
DIFFERENT           > 
        version1.txt< This is line one
           count.txt> This is line 1
DIFFERENT           > 
        version1.txt< This is second line
           count.txt> This is secont line
MATCH               = This is third line

===============
[download]

Oh, right, aside from the comments I made, I also added newlines to your print statements.