Historically this was to help the compiler. You had to give it the list of names before it used them - whether this was the actual usage, or a forward declaration (C's default funcion prototype aside).
Modern compilers for modern languages show that this is no longer a necessity, so C & C++'s (as well as Objective-C, and probably others) syntax here is histotical baggage. In fact one this is one of the big problems with C++ that even the addition of a proper module system will not solve.
Disadvantages are: lots of heavily nested include files (I've traced include trees before, they are surprisingly huge) and redundancy between declaration and definition - all leading to longer coding times and longer compile times (ever compared the compile times between comparable C++ and C# projects? This is one of the reasons for the difference). Header files must be provided for users of any components you provide. Chances of ODR violations. Reliance on the pre-processor (many modern languages do not need a pre-processor step), which makes your code more fragile and harder for tools to parse.
Advantages: no much. You could argue that you get a list of function names grouped together in one place for documentation purposes - but most IDEs have some sort of code folding ability these days, and projects of any size should be using doc generators (such as doxygen) anyway. With a cleaner, pre-processor-less, module based syntax it is easier for tools to follow your code and provide this and more, so I think this "advantage" is just about moot.
与恶龙缠斗过久,自身亦成为恶龙;凝视深渊过久,深渊将回以凝视…