JWT: Choosing between HMAC and RSA
HMAC is used to protect against manipulation by someone who has not access to the secret. Typically this means to protect against manipulation by the client if the secret is only known to the server or to protect against manipulation in transit if the secret is known to client and server.
An RSA based signature is used to protect against manipulation and also to allow others to verify the integrity and source of the data without being able to manipulate the data. This can be done because the private key is only known to the provider and signer of the data while the public key is known to everybody who likes to verify the integrity.
What to choose thus depends on your specific use case. If the server just needs to protect the token against manipulation by the client then using HMAC with a server side secret is enough. If instead it needs to be also proven to others that the token was created by a specific (trusted) server then RSA based signatures should be used.