...I take in a text file (words.txt) where there is a new word each line.

If each line is one word, splitting isn't necessary. Adapting your script, you can do the following:

#!/usr/bin/env perl
use strict;
use warnings;

open my $words, '<', 'words.txt' or die $!;

while ( my $word = <$words> ) {
    print $word;
}

print "\n\n";

close $words;
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You may have noticed from the excellent responses that lexical variables (my) are used. It's best to follow this practice--even with file handles. Also, always use strict; and use warnings;. It's also best to handle open errors, in case there was a problem opening a file.

Hope this helps!