Why does Apple differentiate between new and replacement? M vs N. To serve what purpose?

As it shows….to differentiate if the device is a brand new retail product (if I am buying something new I want it to be completely new…ie a retail unit and not a like new refurbished one or one that was never intended to be sold and is only in inventory for customers needing a warranty replacement). The designations clearly identify what the device is. The only people that I have noticed tend to dislike the fact that there is a designation that is identifiable between a new device and one that is a replacement for a previously owned device that had to be replaced under warranty is someone that is trying to sell the replacement device as new.

I think you answered your own question. Each designation is quite clear, wouldn't you agree? M is a new phone, never used before. F and N are replacement units, which are generally refurbished devices. And P is engraved. I'm not sure what isn't clear in the designations? And the designation would be contained in the device Serial number as it should be. Don't you think if someone is looking at a used iPhone, they should be able to know by the serial number if was sold as New or Refurbished? I'd certainly appreciate know this if I were to ever consider buying a used phone.

Why does Apple differentiate between new”M”, and replacement “N”?

Because a replacement is not new.