It seems to me the biggest problem with the composite bow was its complex construction. Thanks to its many parts and the organic glue that held it together I imagine the composite bow didn't fare as well as the longbow in adverse weather conditions (I wonder how those Mongol bows fared in Vietnam's sultry, jungle rot inducing climate) and I'm guessing it didn't take well to being mishandled or banged around. In contrast a longbow was remarkably simple in its construction as it was made from a single piece of wood with only a few extra bits used for the nocks. Not that the longbow was an all-weather wonder but it's single piece construction seems to naturally endow it with greater constitution and longevity.

After reading the link provided by econ21 I was surprised to learn that keeping bowstrings dry wasn't nearly the pain in the rear we've been led to believe...

Bow strings were of two materials: in the sixteenth century, strings were made of "good hempe...(but, earlier, strings were made of)...fine Flaxe or Sylk".47 A waterproof glue was used to preserve the Renaissance bow string and it was reinforced by a whipping of fine thread.48 The strings were attached to nocks made of bone or horn.49