Also, you used max_by even in the second paragraph where you wanted min_by.

After fixing these, it works as you expected:

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

use List::UtilsBy qw{ min_by max_by };

my %counts = (
    Adam => { "201708" => 1,
              "201703" => 1,
              "201804" => 1,
              "201603" => 1,
              "201705" => 1,
              "201702" => 1,
              "201608" => 1,
              "201704" => 1,
             },
    Sam  => { "201803" => 1,
              "201801" => 1 },
 );

for my $name ( keys %counts ) {
    my $subhash = $counts{$name};
    my $maximal = max_by { $_ } keys %$subhash;
    print "$name, $maximal\n";
}


for my $name ( keys %counts ) {
    my $subhash = $counts{$name};
    my $minimal = min_by { $_ } keys %$subhash;
    print "$name, $minimal\n";
}
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Update:

I also moved the use statement to the top (as it's executed during the compilation phase, anyway) and replaced foreach with its shorter form.

Note that you might use the minmax_by function, too:

use List::UtilsBy qw{ minmax_by };

for my $name ( keys %counts ) {
    my $subhash = $counts{$name};
    my ($minimal, $maximal) = minmax_by { $_ } keys %$subhash;
    print "$name, $minimal - $maximal\n";
}
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Update 2: But using { $_ } means you can get back to List::Util::min or max, or List::MoreUtils::minmax.