There is a weak security property that RSAES-OAEP + AES-CCM (or any KEM + authenticated encryption) provides that RSAES-OAEP + AES-CBC (or any KEM + unauthenticated encryption) does not: Without authentication, a MITM on the channel from sender to receiver can make a small tweak to a message that they otherwise don't know much about; with authentication, the MITM can only replace the message wholesale.
Generally, it is better to just use authenticated encryption everywhere by default unless you have an overwhelming performance constraint and a security justification for avoiding authentication. This is why ‘DEM’ as the modern approach to public-key encryption—which is how all the NIST PQCRYPTO submissions work, for instance—is defined to be the composition of KEM with authenticated encryption.
Don't think of it as AES in CBC mode. Think of AES-CBC as a tool with a security contract—and a confusing one, at that; you are better off with the security contracts of authenticated encryption tools like AES-GCM (also detailed at that link) or AES-CCM or, better, NaCl crypto_secretbox_xsalsa20poly1305.
Finally, if you want public-key authenticated encryption, where the sender and receiver both share their public keys in the telephone book, and the sender sends a message that only the receiver can open and that the receiver can verify came from the sender, then you should just use a public-key authenticated encryption scheme like NaCl crypto_box_curve25519xsalsa20poly1305.
Note that public-key authenticated encryption is different from the composition of public-key signature and public-key anonymous encryption: first, generic composition of them is tricky; second, a public-key signature is verifiable by anyone, not just the receiver, which means a malicious receiver can present the signed to a third party who also knows the sender's public key. With mere public-key authenticated encryption, the third party has no cryptographic assurance the receiver didn't just forge the putative message, because only the receiver can verify messages.