Date::Calc should do the job. For your first situation with dates in year/month/day format:

my $string_one = "02/03/08"; # assuming year/month/day
my $string_two = "02/04/02";

my @date_one = split /\//, $string_one;
my @date_two = split /\//, $string_two;

$date_one[0] = 20 . $date_one[0];
$date_two[0] = 20 . $date_two[0];

my $time_one = Date_to_Time(@date_one, 0, 0, 0); # 0s are for hour/min
+/sec
my $time_two = Date_to_Time(@date_two, 0, 0, 0);

if ($time_one > $time_two) {
    print "$time_one is newest.\n";
} elsif ($time_one < $time_two) {
    print "$time_two is newest.\n";
} else {
    print "Same date (to the nearest day).\n";
}
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Comparing dates in hour:minuteAM/PM should be simple as well, just split the string for the hour, minute, and AM/PM parts, add 12 to the hours if it contains PM, convert to minutes, and compare.