In Hearthstone if you're losing a game you can simply concede and I was wondering if there is any logic behind it other than making an obvious loss end quicker. Does either player gain a little less XP or does just the conceder gain less XP, etc.?
Conceding in Hearthstone, as with any other game, causes you to immediately lose and end the game. It's for when you have no way of winning and allows you to move onto your next game without waiting for the inevitable.
Neither player is penalized for this, though players will earn less xp than they would by finishing the game as xp is (either directly or indirectly) based on the length of the game.
In Hearthstone, if you concede, there are a few losses that can be incurred for one or the other. The first, as mentioned in Studoku's answer, is that the amount of xp will be a bit less, due to the length of a game being a factor of xp earned. In addition, if one player or the other has a quest such as: "Deal X Damage to your opponents." or "Play x 2 cost or lower minions", one player conceding might cause the player with this kind of quest to not progress quite as far in their quest as if you had just played it out. For example, often times if I'm going to win, and I have a quest like this, I will typically play out the winning turn such as playing a couple more 2 cost or lower minions, or playing a few more spells if doing so will advance my quest, rather than just killing my opponent at the start of the turn.