I looked up the implementation and it's even more mysterious:
-- | Sequence actions, discarding the value of the first argument.
(*>) :: f a -> f b -> f b
a1 *> a2 = (id <$ a1) <*> a2
-- This is essentially the same as liftA2 (flip const), but if the
-- Functor instance has an optimized (<$), it may be better to use
-- that instead. Before liftA2 became a method, this definition
-- was strictly better, but now it depends on the functor. For a
-- functor supporting a sharing-enhancing (<$), this definition
-- may reduce allocation by preventing a1 from ever being fully
-- realized. In an implementation with a boring (<$) but an optimizing
-- liftA2, it would likely be better to define (*>) using liftA2.
-- | Sequence actions, discarding the value of the second argument.
(<*) :: f a -> f b -> f a
(<*) = liftA2 const
I don't even understand why does <$ deserve a place in a typeclass. It looks like there is some sharing-enhancig effect which fmap . const might not have and that a1 might not be "fully realized". How is that related to the meaning of Applicative sequencing operators?