It should suffice to say whether bcrypt or SHA-512 is good enough. And the answer is yes, either algorithm is secure enough that a breach will occur through an implementation flaw, not cryptanalysis.
If you insist on knowing which is "better", SHA-512 has had in-depth by NIST and others. It's good, but flaws have been recognized that, while not exploitable now, have led to the the SHA-3 competition for new hash algorithms. Also, keep in mind that the study of hash algorithms is "newer" than that of ciphers, and cryptographers are still learning about them.
Even though bcrypt as a whole hasn't had as much scrutiny as Blowfish itself, I believe that being based on a cipher with a well-understood structure gives it some inherent security that hash-based authentication lacks.
Note: bcrypt is an algorithm that uses Blowfish internally. It is not an encryption algorithm itself. It is used to irreversibly obscure passwords, just as hash functions are used to do a "one-way hash".
Cryptographic hash algorithms are designed to be impossible to reverse. In other words, given only the output of a hash function, it should take "forever" to find a message that will produce the same hash output. In fact, it should be computationally infeasible to find any two messages that produce the same hash value. Unlike a cipher, hash functions aren't parameterized with a key; the same input will always produce the same output.
If someone provides a password that hashes to the value stored in the password table, they are authenticated. In particular, because of the irreversibility of the hash function, it's assumed that the user isn't an attacker that got hold of the hash and reversed it to find a working password.
Now consider bcrypt. It uses Blowfish to encrypt a magic string, using a key "derived" from the password. Later, when a user enters a password, the key is derived again, and if the ciphertext can be decrypted successfully with that key, the user is authenticated. The ciphertext is stored in the "password" table, but the derived key is never stored.
In order to break the cryptography here, an attacker would have to recover the key from the ciphertext. This is called a "known-plaintext" attack, since the attack knows the magic string that has been encrypted, but not the key used. Blowfish has been studied extensively, and no attacks are yet known that would allow an attacker to find the key with a single plaintext.
So, just like irreversible algorithms based cryptographic digests, bcrypt produces an irreversible output, from a password, salt, and cost factor. Its strength lies in Blowfish's resistance to known plaintext attacks, which is analogous to a "first pre-image attack" on a digest algorithm. Since it can be used in place of a hash algorithm to protect passwords, bcrypt is confusingly referred to as a "hash" algorithm itself.
Assuming that rainbow tables have been thwarted by proper use of salt, any truly irreversible function reduces the attacker to trial-and-error. And the rate that the attacker can make trials is determined by the speed of that irreversible "hash" algorithm. If a single iteration of a hash function is used, an attacker can make millions of trials per second using equipment that costs on the order of $1000, testing all passwords up to 8 characters long in a few months.
If however, the digest output is "fed back" thousands of times, it will take hundreds of years to test the same set of passwords on that hardware. Bcrypt achieves the same "key strengthening" effect by iterating inside its key derivation routine.
So, my recommendation of bcrypt stems from the assumptions 1) that a Blowfish has had a similar level of scrutiny as the SHA-2 family of hash functions, and 2) that cryptanalytic methods for ciphers are better developed than those for hash functions.