If you try to design an algorithm, what you get is a model of how you think your system behaves. If you train a neural network, what you get is a model of how your system actually behaves. In other words, a neural network does not embed any assumptions or preconceived notions of how the system ought to work. Intuitively, not incorporating knowledge of how the worlds works ought to be a disadvantage, but in practice it turns out to be not just an advantage, but an embarrassingly large one.

Neural Networks aren't the only common approach in modern AI research, but the other techniques (e.g. Support Vector Machines) have similar properties: they're dumb methods of finding correlations in data. The field of AI is slowly coming to grips with the fact that more training data trumps better algorithms.

That said, I sympathize with your frustrations. You can solve a lot of problems by throwing more cycles at it, but you don't learn anything, either about the algorithms or about your target domain. For example, a human plays the game of Go differently than a machine. A human has an intuition of what moves are good and what are bad, and only considers a handful; a machine relies less on evaluating board position and more on brute-forcing through possible board configurations. As machines get more powerful, eventually the machines catch up to humans in ability, with that same approach. While their methods will become impressive in their efficacy, we learn nothing about how humans are so much better at evaluating board position and selecting moves to speculate on.