This is a simple but very useful example of a problem that can be quickly solved using Tie::File. If you tie both files to their own arrays, searching and deleted between the two becomes easy.

use strict;
use Tie::File;

tie my @file1, 'Tie::File', 'file1.txt' or die;
tie my @file2, 'Tie::File', 'file2.txt' or die;

@file1 = grep { my $name = $_; ! grep {$_ eq $name} @file2 } @file1;
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That last line iterates over each line of @file1 and only lets it hang around if it's not found in @file2. Using your example files, the code above transforms file1.txt into the follow:

nick
peter
adam
wolfgang
antony
mike
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See node 629253 for links to tutorials on Tie::File.