What would be the "distinct method" that Traversable has in addition to Foldable?

Foldable is to Functor as Traversable is to Monad, i.e. Foldable and Functor are superclasses of Monad and Traversable (modulo all the applicative/monad proposal noise).

Indeed, that's already in the code

instance Foldable f => Traversable f where
  ...

So, it's not clear what more there is to want. Foldable is characterized by toList :: Foldable f => f a -> [a] while Traversable depends ultimately on not only being able to abstract the content as a list like toList does, but also to be able to extract the shape

shape :: Functor f => f a -> f ()
shape = fmap (const ())

and then recombine them

combine :: Traversable f => f () -> [a] -> Maybe (f a)
combine f_ = evalStateT (traverse pop f_) where
  pop :: StateT [a] Maybe a
  pop = do x <- get
           case x of
             [] = empty
             (a:as) = set as >> return a

which depends on traverse.

For more information on this property see this blog post by Russell O'Connor.

Super hand-wavy because it's late, but the extra power that Traversable has over Foldable is a way to reconstruct the original structure. For example, with lists:

module MyTraverse where

import Data.Foldable
import Data.Traversable
import Control.Applicative
import Data.Monoid

data ListRec f x = ListRec
  { el :: f (Endo [x])
  }

instance Applicative f => Monoid (ListRec f x) where
    mempty = ListRec (pure mempty)
    mappend (ListRec l) (ListRec r) =
        ListRec (mappend <$> l <*> r)

toM :: Functor f => f b -> ListRec f b
toM this = ListRec $ (Endo . (:)) <$> this

fromM :: Functor f => ListRec f b -> f [b]
fromM (ListRec l) = flip appEndo [] <$> l

myTraverse :: Applicative f => (a-> f b)  -> [a] -> f [b]
myTraverse f xs = fromM $ foldMap (toM . f) xs

I think this myTraverse behaves the same as traverse, using only the classes Applicative, Foldable, and Monoid. You could re-write it to use foldr instead of foldMap if you wanted to get rid of Monoid.

lists are easy because they're a flat structure. However, I strongly suspect that you could use a Zipper to get the proper reconstruction function for any structure (since zippers are generically derivable, they should always exists).

But even with a zipper, you don't have any way of indicating that structure to the monoid/function. Notionally, it seems Traversable adds something like

class Traversed t where
  type Path t :: *
  annotate :: t a -> [(Path t, a)]
  fromKeyed :: [(Path t, a)] -> t a

this seems to overlap heavily with Foldable, but I think that's inevitable when trying to associate the paths with their constituent values.