Is there any reason I should aspire to play against and defeat the Expert Practice AI? Does it give me more exp, or gold or some other form of compensation?
Expert mode differs from basic mode in that the expert mode heroes will use "a large number of expert cards, including many common cards, a few rares, and the odd epic, but do not include any legendaries."
Defeating all expert heroes will complete the Crushed Them All! achievement quest, awarding 100 gold; progress towards this achievement is not tracked anywhere. Aside from Crushed Them All! player wins in Expert Mode earn experience towards hero levels (slightly less than playing humans) and if the quest is currently assigned count towards the Total Dominance Daily Quest.
The Practice Mode article on Gamepedia provides additional details about the differences between the two modes.
Some good answers are here already, but I wanted to add another benefit of playing against the expert AI (as opposed to human opponents): There's no turn time limit. If you want to get some practice in while doing something else that might draw you away from your computer for minutes at a time (cooking, caring for a child, etc.), then playing against the AI is decidedly less stressful.
The point is to practice against the expert AI, and hone your hearthstone skills.
In this mode, players can hone their abilities and test out new deck ideas against A.I. opponents of basic or expert skill levels, and unlock new playable heroes in the process. Expert skill setting is only available after all the heroes are unlocked.
You won't be earning any real gold or any playable heroes in the Expert Practice AI, but you can test new strategies against opponents and see how the AI reacts against them. It's just for practice.