Aside from the first win, are there any rewards to playing Tavern Brawl?
Solution 1:
No, it gets the same treatment as casual mode in regards to rewards. As in it counts towards daily quests and you get the usual 10 gold per 3 wins.
Solution 2:
Nope. Currently there are no rewards beyond the first win.
I speculate that in the future there might be rewards for more wins or alternatively you will have to win multiple times to get the reward.