What is the advantage of arrows over regular functions in haskell. What can they do the functions can't. Functions can map over structures using fmap.
On more of a broad picture, arrows get you out of Hask and into other categories there are to explore. The Kleisli category is probably the best-acquainted to Haskellers, followed by Cokleisli. Those are the natural "extensions" of Hask: add an endofunctor around either the result or argument, then you get a category again if
Kleisli: the functor is a monad, so id ≅ return :: a -> m a
(.) ≅ (<=<) :: (b->m c) -> (a->m b) -> a->m c
CoKleisli: the functor is a comonad, so id ≅ coreturn :: m a -> a and
(.) :: (m b->c) -> (m a->b) -> m a->c
(For that you don't need Arrow yet, only Category. But general categories aren't very interesting, you normally want monoidal or even cartesian closed categories, which is what Arrow is roughly aiming at.)
But there are sure are lots of other categories. Most don't have much to do with Hask and can't be expressed with the standard Arrow class, mainly because the objects have special properties that not every Haskell type fulfills. Actually, if you add the ability to constrain the object types, the possibilities immediately become much wider. But even if you stay with the standard classes, perhaps even simply in ->, the point-free composition-style that is natural with arrows often comes out very nice, concise, and opens up new ways to think about transformations.