The first playthrough of the normal mode bosses and the class challenges gives you new cards for your deck(s).
And the first complete playthrough on heroic mode gives you a new card back.
But is there any reason to replay the new solo adventure after you have cleared it one time?
No, you’ll just earn 6 cards (1 legendary and 5 ‘commons’).
Cards you’ll earn :
Haunted Creeper good for a zoo deck (it’s a deck with low cost cards and very agressive) you’ll earn this if you beat Anub’Rekhan in normal.
Nerub’ar Weblord good for counter battlecry, you’ll earn this if you beat Faerlina in normal.
Nerubian Egg it’ll combo with a solfurie or an Argus, because if you destroy it, you’ll earn a nerubian (it’s a 4/4 cards), you’ll earn this if you beat Maexxna in normal.
Maexxna she destroy any card she attacked, it can be usefull if you play Anduin Wrynn because he can heal Maexxna. You’ll earn this if you beat all the 3 boss in normal and you’ll unlock heroic and druid and rogue class challenges.
Poison Seeds : destroy all monsters on the battlefield and give treants 2/2 for each destroyed monster. Can combo with a druid token. Unlock when you finish the druid class challenge.
Anub’ar Ambusher a 5/5 for 4 mana but with a Deathrattle (Return a random friendly monster in your hand) can combo with Si 7. Unlock when you finish the rogue class challenge.
You can find many others information on the wiki : http://hearthstone.gamepedia.com/Curse_of_Naxxramas