I'm currently studying modern public key cryptography course, and I'm confused whether public key and asymmetric cryptography are the same.
Yes, they are the same in most contexts. The asymmetry is in the keys used, public and private keys. It's definitely not in the encryption and decryption method; in (textbook) RSA the method for encryption and decryption (modular exponentiation) is completely the same. RSA is certainly used for public key cryptography and it is certainly also an asymmetric algorithm.
Key agreement and signature generation are also asymmetric cryptography. You could say that "public key cryptography" puts some more emphasis on the public key, probably hinting that the public key is generated by one party and trusted by another. This is usually the case for PKI(X), the underlying certificate based public key infrastructure used for e.g. TLS / HTTPS.
The public & private key may however not be static or trusted for Diffie-Hellman key agreement where the public / private key pairs are ephemeral (generated per key agreement operation).
Diffie-Hellman however also works with static/trusted keys, so that difference is feeble at best.
- Asymmetric private keys are sometimes called secret keys. I think private keys is the better term. I use the term "secret keys" for symmetric keys (those may not be private to one party but must be kept secret by the parties holding them).