Advantage: Fast switching. Switching between fibers is done by swapping in new values for the program counter and stack pointer. Compare with async, where yielding only takes you up one level of stack, so your code ends up walking a stack of coroutine invocations to get all the way back to the scheduler.
Disadvantage: Expensive setup - each fiber needs its own pre-allocated stack.
Disadvantage: Operating systems don't provide I/O systems readymade to work with fiber schedulers, so if you want to yield to a different fiber during blocking I/O, then you need to invent your own system to handle that.
PS about naming: "Fiber" is the Microsoft Windows name for their implementation of full stack coroutines, but the proper platform-independent name for it is actually "coroutine". I've kept with the word "fiber" here to avoid misunderstandings.
PS about C++: This is based on async like you find it in C# or Python, I don't really know about C++ coroutines.