You don't do any error checking and you don't use warnings which would allow Perl to alert you of unopened files. When opening files, use the following:

...
open (FILE1,$p)
    or die "Couldn't open file '$p': $!";
...
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Also, your approach will be very (very) slow, because for each entry in the first file you will go through all lines of the second file. It will likely be better to store the entries from the smaller file in a hash and then go through the larger file once and compare the entries to it.