Should often gives advice, while have to expresses necessity or obligation. Don’t have to means something is not necessary, not that it is prohibited.
Practical examples
You should check the address.
من الأفضل أن تتحقق من العنوان.
I have to finish this today.
يجب أن أنهي هذا اليوم.
You do not have to bring food.
لست مضطرًا إلى إحضار الطعام.