请始终尝试使用debug()函数作为变量，以了解两者之间的区别。

debug($appointment_time1);
debug($appointments);

在你的例子中，$appointment_time不是你所期望的，而是\Cake\ORM\Query对象，而$appointment_time1是一个字符串。

试试看：

// convert posted string to date time format
$appointment_time1 = date('Y-m-d H:i:s', strtotime($this->getRequest()->getData('datetime')));

// fetch data from db
$appointments = $this->Appointments->find()->select(['appointment_time']);

foreach ($appointments as $appointment) {
   echo $appointment->appointment_time->format('Y-m-d H:i:s') == $appointment_time1 ? 'OK' : 'NOT OK';
}

或

$appointment = $this->Appointments->find()->select(['appointment_time'])->first();

echo $appointment->appointment_time->format('Y-m-d H:i:s') == $appointment_time1 ? 'OK' : 'NOT OK';

还可以尝试：

$appointment->appointment_time->getTimestamp() == strtotime($this->getRequest()->getData('datetime'));