Actually, that wikipedia article already answers your question:
It is moderately common for companies and sometimes even standards bodies as in the case of the CSS encryption on DVDs – to keep the inner workings of a system secret. Some argue this "security by obscurity" makes the product safer and less vulnerable to attack. A counter argument is that keeping the innards secret may improve security in the short term, but in the long run only systems that have been published and analyzed should be trusted.
In the end, Kerckhoffs' surviving principle nr. 2 (of initially six design principles for military ciphers, which published in 1883) says nothing else than that you have to look at the security of your crypto under the aspects of "the enemy knows the algorithm" and maybe even "the enemy carries the message". The well-known Claude Shannon once simplified Kerckhoffs' principle as: "the enemy knows the system".
The "advantage to the attacker" you are thinking about would practically be "just a matter of time for the attacker to find out" since keeping the algorithm "secret" would be relying on "security through obscurity", which we all know to be a bad idea.
The idea of Kerckhoffs' principle is that if any part of a cryptosystem (except the individual secret key) has to be kept secret then the cryptosystem is not secure. That's because if the simple act of disclosing some detail of the system were to make it suddenly insecure then you've got a problem on your hands. The gold standard for any secret keeping system is that all its details should be able to be made public without compromising the security of the system. The security relies on the system itself, not the secrecy of the system.
A good example of why Kerckhoffs' principle makes sense would be the German Enigma machine used in WWII. By stealing machines, the Allies knew everything there was to know about how the Enigma machine worked; they only had to read the manuals, use the code-books and look at the way the machine was constructed and worked. After all, they had a working example in their hands. Throughout history, there are countless examples of secret algorithms falling into wider knowledge, variously through espionage, betrayal and reverse engineering. And on occasion, ciphers have been reconstructed through pure deduction. For example, the German Lorenz cipher and the Japanese Purple code, and a variety of classical schemes.
Therefore, Kerckhoffs' principle is still applied today. The security of HTTPS, SSL and ciphers like AES or RSA rely on the complexity of the algorithm, not on keeping them secret. They algorithms are all published, detailed standards which everyone can take a look at. The only "secret" is the key that's chosen when you connect to a secure web site (that's done automatically and randomly by your browser and the server) or when you encrypt a document using a program like GPG.
So why respect Kerckhoff's princible? Because it's "just a matter of time for the attacker to find out" how your crypto works. Security gain can therefore only be of temporary nature. Also, publication builds trust.