Is there a difference between an NFC and RFID chip?
NFC is a type of RFID. Not all RFID devices use NFC.

NFC is a bundle of specific technologies and protocols used to communicate with storage devices (often credit cards or passive identifier tags). The NFC specifications cover everything from the radio frequency used (13.56 MHz) and the types of modulation used, to the communications protocol used to read and write data on tags.

RFID, by contrast, is a generic term used to refer to any form of radio communications with an identifier tag. There are dozens, if not hundreds, of different systems which fall under this category, and most of them are mutually incompatible. NFC is often considered a type of RFID, but they are not synonymous.

Pet microchips do not use NFC. There are a variety of different RFID systems used for these chips; most of them operate at 125, 128, or 134.2 kHz, and they are largely incompatible with each other. (It is not unheard of for an animal to be implanted with multiple brands of ID microchips to increase the odds that they will be identified!) A NFC reader in a cell phone will have no way of communicating with these chips.