The easiest way to distinguish a quest from an encounter is that quests don't involve just a single encounter, but are a goal completed through a series of encounters.
While you may have a quest to kill a dragon, you wouldn't have a guy outside the dragon lair saying "Defeat this dragon and I'll reward you!", followed by a single fight with a dragon. Instead, you might have a village who, out of desperation, pleads with the party to defeat the evil dragon preying on their livestock. Thus, the quest could possibly involve learning more about the dragon, locating the dragon's lair, making your way there, defeating its minions, and killing it (or negotiating with it, or driving it away).
Ideally, quests can be completed in more than one way, and don't necessarily have to be completed in a set order, or in a set sequence.
Quests are useful to keep a party focused on a longer term goal (rather than just fight this, what's next?). They are helpful to encourage players (and characters) to learn more about the setting, and to explore and interact with things in more than just a combat encounter setting.
In addition, they can be used to broaden out a story line (everybody knows we are here to kill the dragon, but since we talked to the priest, we know that a treasured relic is part of the hoard, so we need to look for it and return it to the temple). In addition, they are very useful to move the story forward when there is no encounter-driven push.
For instance, I'm running Pyramid of Shadows. I used the "Bring me the head of Vren" quest to drive the party towards the pyramid, even though it, in itself, has nothing to to with the bandit chief they are hunting. The quest is incidental to the adventure, but it was a hook to get them into the adventure. That quest also gave them some xp, and motivated their search through the pyramid ("While we're here, let's look for the bandit that was supposedly trapped too).
Now that they are in the Pyramid, they had the quest to recover the life force of Vyrellis. This was a useful way to drive them forward into exploring areas that they might have avoided (let's just bypass this dragon, shall we?), and allowed them to succeed by negotiation, or theft, or combat.
Then I took that quest and altered it, in that they need to take the recoverd orb and return it to her kinfolk in the Feywild. That part of the quest (taking it back to the Moonstair) will be totally easy once they escape, and won't even require any skill challenges (the challenge was restoring her life force and escaping in the first place). However, it will drive the party from the current adventure into the next one, even though it doesn't have any connection.
Quest cards are useful for many reasons, but the one I find most important is simply that you probably play once a week, or even less often, with your group. While, in the game, the adventure may only take a few days, or even less, during the sessions, it might stretch over months. Things that would be easily remembered to the characters are often quite easily forgotten by the players. The quest cards are a way for the story goals to be in the forefront of the player's minds.