Thinking about this from a Haskell perspective, it seems that with promises the lambda waiting for the callback is basically encapsulated into the promise:

  type Promise a = (a -> IO ()) -> IO ()
  getData :: Promise String

Whereas with callbacks you pass that around explicitly:

  getData :: (String -> IO ()) -> IO ()

Promises are a monad, so you get the usual monad advantages of being able to "hide the plumbing" behind the abstraction, hence no tower of explicit callbacks in your JS code.

  instance Monad Promise where 
    return x = \k -> k x
    fmap f p = \k -> p (k . f)
    mx >>= fxmy = \k -> mx (\x -> fxmy x k)