A few days ago I posted suggesting that you salt your passwords, I'm back armed with even more knowledge and better advice. Turns out the relative strengths of one hashing algorithm vs another can in fact make a difference, in a way I didn't even consider - their speed.
Most crypto hash functions are designed for speed, you want to be able to compute the hashes of lots of data pretty quickly if your pushing it down the wire. That speed works in an attackers favour if they're brute forcing a list of passwords and newer hashing functions can make it worse, one of the requirements for SHA-3 is that it's faster than the SHA-2 family.
So what's the new right answer?
Choose a function that takes enough time that an attacker has to work for each and every password - ideally long enough that it would take forever to crack just one - while making sure that legitimate users aren't waiting forever while you check their passwords.
There are two ways of doing this, run a fast hash function many times or deliberately pick a slow hash function.
Running many iterations of a fast hashing algorithm is pretty self explanatory, run it twice and it takes twice as long, run it a thousand times and it takes a thousand times as long to attack each password.
Bcrypt is an example of the second, based on the blowfish algorithm it uses the fact that the key setup step is a relatively expensive operation and difficult to optimise. By making use of this bcrypt allows you to set a work factor and creates a hashing algorithm that is expensive and also difficult to optimise.
Which is better? I have no idea, both are widely used and it really depends on your environment. I'd love to hear what others think though.