No, plain functors, applicative functors, and monads each deal with a different potential use case.

Look at the specific type signatures:

  plain functor - fmap: (a -> b) -> (F a -> F b)
  applicative functor - ap: A (a -> b) -> (A a -> A b)
  monad - bind: (a -> M b) -> (M a -> M b)

So yes, monads are not unique in doing wrapping and unwrapping data to use with functions--all functors do this--but all monads are also functors, and also applicative functors, so they do everything the other two do, and more. You can't get at anything like bind using just fmap, ap, and pure/unit. But you can write fmap and ap merely in terms of bind and unit.