Welcome to the monastery! You're doing a few things that are just considered bad convention:

• Don't use bare word file handles. Prefer lexical file handles instead. open(my $fh, "<", "/path/here/") or die "Can't open foo: ", $!;
• Use 3-arg open. Please take a look at perlopentut and the example I quoted above. Using 2-arg open, leaves you vulnerable to security issues.
• This is a minor issue - have you considered passing in arguments to your subroutine, instead of reading them from STDIN? Passing to subroutine makes it clear what it's doing, besides keeping the whole logic self-contained.

Good on you for posting the code and I can see that you're doing a lot of work, which would just as easily achieved by using an available module such as File::Compare. File::Compare is bundled with standard perl distributions and should be available with your local installation. Sure, it doesn't print the exact differences which is what you wanted to do, but you can copy steal ideas from it, surely? (For example, instead of reading line by line, you can directly read into a buffer on some configurable buffer size using read or sysread functions?)

As always, have fun! :-)