<p>Generally speaking what do you get out of extending an artificial neural net by adding more nodes to a hidden layer or more hidden layers?</p>

<p>Does it allow for more precision in the mapping, or does it allow for more subtlety in the relationships it can identify, or something else?</p>

This is very interesting question but it's not so easy to answer. It depends on the problem you try to resolve and what neural network you try to use. There are several neural network types.
I general it's not so clear that more nodes equals more precision. Research show that you need mostly only one hidden layer. The numer of nodes should be the minimal numer of nodes that are required to resolve a problem. If you don't have enough of them - you will not reach solution.
From the other hand - if you have reached the number of nodes that is good to resolve solution - you can add more and more of them and you will not see any further progress in result estimation.
That's why there are so many types of neural networks. They try to resolve different types of problems. So you have NN to resolve static problems, to resolve time related problems and so one. The number of nodes is not so important like the design of them.