You use 'complain ABOUT' in the context of making it clear that you are dissatisfied with something: "He's complaining about the state of the room again...." "They are complaining about the noise..." You use complain OF mainly in the context of ailments or illnesses "She's been complaining of headaches for the past three months..." "He's been complaining of nausea since he ate those dodgy mussels...."

The MacMillan Dictonary examples provided by Chris are, of course, correct but I my understanding is that "to complain of" is usually complaining about something that concerns you more directly, personally and intimately. As Chris and Jane have both said, one can complain of an ailment (such as arthritis or gout). Or, as in the case of the MacMillan example, people might complain of being robbed. If you complain about something, that something might be a bit more detached from you and not affect you quite so directly or personally. Here's a couple of examples: People in Seoul often complain about the air pollution but don't feel empowered to do anything about it. Within a few days of arriving in Seoul, the tourist complained of having difficulty breathing caused by the air pollution in the city. Visitors to (pick your favorite dangerous city) complain about the need to be on the lookout for pickpockets and thieves. My friends complained of having been robbed at gunpoint in broad daylight. Hope that helps.

When you "complain about" something, it means you made a statement -- often in detail -- of the reasons that you are dissatisfied with something or someone. When you "complain of" something, it is merely a statement that something has happened, or that you have an ailment. Please see this link about "complain of": http://www.macmillandictionary.com/dictionary/british/complain-of

You say "complain about" + [something that is not always bad - there is just a bad aspect to it on this occasion] "He complained about the food" (food is not always bad) You say "complain of" + [Something is always understood to be bad] "He complained of problems with the food" (problems are always bad)

complain about (something specific) to complain (general) You can complain about your wife's cooking >>> "I don't like this dish". You like to complain. >>> "I don't like this, I don't like that, I don't like anything". You generally just like to complain.