NTAG203 VS NTAG213
These two NFC Chips are very similar: they belong to the same family (NTAG®) and both have universal compatibility with all NFC smartphones and tablets.
Specifically, however, the NTAG213 chip is the new version of the NTAG203: it has more memory and some more functions, such as the scan counter and the password lock.
Currently, the NTAG203 chip is out of production, precisely because it has been replaced by the NTAG213. If you have developed applications or systems using the NTAG203 as a reference, you can still use the NTAG213 chip without problems.
NTAG213 VS NTAG216
The two chips are both NTAG®, so compatibility is universal with NFC smartphones and tablets. They are both equipped with 7-byte UIDs and support NTAG21x functions (such as password lock and scan counter).
The substantial difference between the two chips is the available memory: the NTAG213 has 144 bytes available, while the NTAG216 has 888 bytes. For this reason, the NTAG216 chip is privileged to contain contacts in V-Card format, with many fields, or when it is required that the chip contains more information.
For the same reason, the NTAG216 chip is more expensive. If you have no particular needs regarding memory, the NTAG213 chip is fine.