PHP check if date between two dates

You cannot compare date-strings. It is good habit to use PHP's DateTime object instead:

$paymentDate = new DateTime(); // Today
echo $paymentDate->format('d/m/Y'); // echos today! 
$contractDateBegin = new DateTime('2001-01-01');
$contractDateEnd  = new DateTime('2015-01-01');

if (
  $paymentDate->getTimestamp() > $contractDateBegin->getTimestamp() && 
  $paymentDate->getTimestamp() < $contractDateEnd->getTimestamp()){
  echo "is between";
}else{
   echo "NO GO!";  
}

Edit: use <= or >= to count today's date.

This is the right answer for your code. Just use the strtotime() php function.

$paymentDate = date('Y-m-d');
$paymentDate=date('Y-m-d', strtotime($paymentDate));
//echo $paymentDate; // echos today! 
$contractDateBegin = date('Y-m-d', strtotime("01/01/2001"));
$contractDateEnd = date('Y-m-d', strtotime("01/01/2012"));
    
if (($paymentDate >= $contractDateBegin) && ($paymentDate <= $contractDateEnd)){
    echo "is between";
}else{
    echo "NO GO!";  
}