JWT public key vs private key signature validation -- what is the difference? - authentication

Question

I am using this library, node-jwks-rsa, to fetch JWT keys from my auth0 jwks.json file in order to verify that the id_token my application retrieves after authentication is actually coming from my auth provider.
Under the hood it uses this method to build a public key PEM
export function certToPEM(cert) {
cert = cert.match(/.{1,64}/g).join('\n');
cert = `-----BEGIN CERTIFICATE-----\n${cert}\n-----END CERTIFICATE-----\n`;
return cert;
}
(Using the x50c as argument from the .jwks file).
which I then use in combination with jsonwebtoken to verify that the JWT(id_token) is valid.
How is this method of verification different from generating a private key(RSA) from the modulus and exponent of the jwks.json file and using it for verification instead? (as example see this library)
Additionally here is function as demonstration that generates a PEM from a mod and exponent (taken from http://stackoverflow.com/questions/18835132/xml-to-pem-in-node-js)
export function rsaPublicKeyToPEM(modulusB64, exponentB64) {
const modulus = new Buffer(modulusB64, 'base64');
const exponent = new Buffer(exponentB64, 'base64');
const modulusHex = prepadSigned(modulus.toString('hex'));
const exponentHex = prepadSigned(exponent.toString('hex'));
const modlen = modulusHex.length / 2;
const explen = exponentHex.length / 2;
const encodedModlen = encodeLengthHex(modlen);
const encodedExplen = encodeLengthHex(explen);
const encodedPubkey = '30' +
encodeLengthHex(modlen + explen + encodedModlen.length / 2 + encodedExplen.length / 2 + 2) +
'02' + encodedModlen + modulusHex +
'02' + encodedExplen + exponentHex;
const der = new Buffer(encodedPubkey, 'hex')
.toString('base64');
let pem = `-----BEGIN RSA PUBLIC KEY-----\n`;
pem += `${der.match(/.{1,64}/g).join('\n')}`;
pem += `\n-----END RSA PUBLIC KEY-----\n`;
return pem;
};
The aforementioned jsonwebtoken library can verify a JWT using either -- but why? If both of these verification methods can validate a JWT signature why do they both exist? What are the tradeoffs between them? Is one more secure than the other? Which should I use to verify most fully?

Answer 1

Using a RSA assymetric key pair, the JWT is signed with the private key and verified with the public. You can not verify a digital signature with the private key
Modulus and exponent are the components of the public key and you can use it to build the public key in PEM format, which is a base64 representation of the public key (modulus and exponent) encoded in DER binary format. You can use PEM, DER or modulus and exponent because the contain the same information
But anybody can't build the private key with modulus and exponent. He would need the private RSA elements, which must be kept secret so that no one can sign for you.