1) It gives you control over your starting hand, which gives some influence over the randomness and adds an extra decision in whether or not you keep the quest.
2) It gives up a card and tempo to get something better later.
3) It gives the opponent information which leads to more counter-play potential.
But then they made all the quests hard to complete and give game-breaking rewards, so most quest decks were filled with cards to complete the quest and then exploit the quest reward. Which meant;
1) You never want to mulligan away the quest because it is so fundamental to everything your deck does.
3) The only counter-play for many quests is 'kill them before they finish the quest'.
That is terrible design.
Then Frozen Throne gave us death knights. However, there were no cards to pull a DK from your deck, which I found to be a massive problem because in slower match-ups you would lose if you drew your DK a few turns after they played theirs. The value over time is insane. This is also true of Jaraxxus but the DKs give you armour and most of the them have an ability to help them stay alive (taunts for druid, sweeper for hunter and priest, weapon for paladin and warrior, stealth for rogue, probably taunts for warlock). I wish there were more risk to playing a DK, or more consistency in drawing them.
Actually, my favourite idea is that the DKs should have been quest rewards. You could no longer throw a DK into any slow-ish deck to give the deck late game power, and slower games would be less dependent on drawing your DK (although there still would be some dependence on the draw for how fast you finish your quest).
Finally, the recruit mechanic was such a stupid idea. There was enough experience to know it could be problematic: look at giants, and patches. Yet Kobolds and Catacombs has now had Call to Arms, Possessed Lackey, and Corridor Creeper nerfed, which shows how difficult it is to balance mana-cheating/recruit cards.