If we compare the types
(<*>) :: Applicative a => a (s -> t) -> a s -> a t
(>>=) :: Monad m => m s -> (s -> m t) -> m t
we get a clue to what separates the two concepts. That (s -> m t) in the type of (>>=) shows that a value in s can determine the behaviour of a computation in m t. Monads allow interference between the value and computation layers. The (<*>) operator allows no such interference: the function and argument computations don’t depend on values. This really bites. Compare
miffy :: Monad m => m Bool -> m x -> m x -> m x
miffy mb mt mf = do
b <- mb
if b then mt else mf
which uses the result of some effect to decide between two computations (e.g. launching missiles and signing an armistice), whereas
iffy :: Applicative a => a Bool -> a x -> a x -> a x
iffy ab at af = pure cond <*> ab <*> at <*> af where
cond b t f = if b then t else f
which uses the value of ab to choose between the values of two computations at and af, having carried out both, perhaps to tragic effect.
The monadic version relies essentially on the extra power of (>>=) to choose a computation from a value, and that can be important. However, supporting that power makes monads hard to compose. If we try to build ‘double-bind’
(>>>>==) :: (Monad m, Monad n) => m (n s) -> (s -> m (n t)) -> m (n t)
mns >>>>== f = mns >>-{-m-} \ ns -> let nmnt = ns >>= (return . f) in ???
we get this far, but now our layers are all jumbled up. We have an n (m (n t)), so we need to get rid of the outer n. As Alexandre C says, we can do that if we have a suitable
swap :: n (m t) -> m (n t)
to permute the n inwards and join it to the other n.
The weaker ‘double-apply’ is much easier to define
(<<**>>) :: (Applicative a, Applicative b) => a (b (s -> t)) -> a (b s) -> a (b t)
abf <<**>> abs = pure (<*>) <*> abf <*> abs
because there is no interference between the layers.
Correspondingly, it’s good to recognize when you really need the extra power of Monads, and when you can get away with the rigid computation structure that Applicative supports.
Note, by the way, that although composing monads is difficult, it might be more than you need. The type m (n v) indicates computing with m-effects, then computing with n-effects to a v-value, where the m-effects finish before the n-effects start (hence the need for swap). If you just want to interleave m-effects with n-effects, then composition is perhaps too much to ask!