Passwords are obscurity, are they not? And in the end, so are 2048 bit RSA keys. It's just a prime-number needle in the haystack. Look enough places/try enough passwords, and you will find it.
Practically speaking, obscurity is a "platform" that lets you bypass everything, whereas knowing a password is more limited since it grants access to a single user. But practically speaking, obscurity and root passwords are similar..
I wonder if there are formal definitions here that makes the separation clear.
The major difference here is this: with security through obscurity, someone can reverse engineer one product and then they've broken all products. This is why someone upstream said "security through obscurity doesn't scale". Security through obscurity is often okay if you're protecting one thing, but if you're using it to protect a system (like a pacemaker) that is going to be used by a lot of people, the more people who use it, the more valuable a reverse engineering hack becomes. Security through obscurity can't be individualized to provide security to each individual--if one system is broken all systems are broken.
Compare this with key-based security--if each instance of the system has an individual, randomized key with a large enough keyspace, breaking a key will only get you into a single instance of the system. It scales because the reward for breaking the security doesn't grow as the number of system instances grows.
Note that the problem with security through obscurity is basically the same problem with master keys, i.e. those used for backdoors or DRM. If someone can obtain the master key for the system, they can break all the instances of the system.
Locks are rated in how many seconds/minutes they can withstand from a dedicated attacker. Perhaps that would be a way to determine a similar safety rating for passwords/crypto based systems.
In passwords: how many passwords can you try per second before the server refuses? Then Password space/# per second = total seconds for guaranteed entry.
Crypto: how many keys do you have to calculate before you succeed in finding the correct key? Keys/second * how many machines / keyspace.
But in the end, I don't believe we really have a formalized difference between types of obscurities, aside the "go to /root and get root" obvious badness. It would be a rather nice way to provide security in "seconds to millenia depending on techniques used".
Most individuals defending algorithmic security through obscurity believe that hiding the algorithm improves security. That may be true in an extremely technical sense (the attacker must recover the algorithm first), but it is very misleading and unprofessional commentary. Algorithmic security through obscurity is at best calculated in difficulty-to-reverse-engineer (or difficulty-to-steal), which doesn't provide per-use(r) specificity (per-user password) nor scale in complexity (a 256-bit key is generally 2^128 times stronger than a 128-bit key, but doubling the algorithm length increases reversing time by slightly less than a factor of 2).
Algorithmic security through obscurity provides negligible security, but what's the harm? Why should we care? Attempting to hide the algorithm provides a false sense of security, limits review to "approved" parties, and induces legal/social efforts to "protect" the secret. The limited review is particularly noteworthy since it promotes bugs in both the algorithm and the implementation. The end result is a facade of security, some very unhappy whitehats, some very happy blackhats, and more users betrayed through poor security practices.
[1] http://csrc.nist.gov/publications/nistpubs/800-57/sp800-57_p... [2] http://tjscott.net/crypto/64bitcrack.htm#INTELG
It is much harder to formalise how hard it is for an attacker to find out what algorithm you use, so it is risky relying too much on him not being able to do so.