Discarding Damage Cards vs Discarding Upgrade Cards
The X-Wing rules reference overrides the Core Rulebook with the following:


  When an Upgrade card is discarded, it is flipped  facedown. The card
  is out of play for all purposes  except when determining the total
  squad point  cost of the ship to which it was equipped. If an  Upgrade
  card is flipped faceup by a game effect,  it returns to play equipped
  to the same ship.


So yes, discarding a damage card does behave differently from discarding an upgrade card.  In part, this is because facedown damage cards already have significant game meaning, where facedown upgrade cards do not.  Some critical damage cards have ways of turning themselves facedown, which is different from discarding them since they remain a damage but no longer have any special effect.